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Rigidity theorem for the equation of characteristics of a second-order linear equation of mixed type on a plane at a point where the coefficients are zero S. M. Voronin, E. A. Cherepanova Chelyabinsk State University, Chelyabinsk, Russia
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  § 1. IntroductionCharacteristics of second-order linear differential equations are described by differential equations of the following form: (see [1]–[3]). Similar equations also arise in other applications (see [4]–[7]) in a natural way. Setting $p={dy}/{dx}$, we can write (1.1) as an implicit differential equation We call equations of this type (as well as ones of the form (1.1)) binary equations (see [5]).Many authors investigated implicit differential equations by considering both the global (see [7]–[10]) and local properties of such equations. Of course, in the local analysis singular points of equations were in the focus. In cases of low codimension local normal forms of implicit equations were constructed in [1] and [11]–[13]; also see [2] and [14]–[18]. These normal forms are rather simple, and they are normal forms for analytic and formal (as well as smooth) classification alike (and can be simplified further for topological classification: see [11]). In cases of codimension $3$ (which are inherent in one-parameter systems of implicit equations) the generic conditions from [11] can fail to hold. In this case, when resonant singular points appear, we can anticipate the same phenomena as for resonant singular points of planar vector fields (namely, a divergence between analytic and formal classifications, when there are functional moduli for the first and numerical moduli for the second). Quite little is known about other violations of conditions of genericity (a tangency between the mirror (that is, fixed point set) of an involution and a separatrix of a singular point: see [11]). Equations with nonsmooth surfaces were considered by Bogaevsky [19] (also see [20]). However, for binary equations, another type of degeneracy is also possible in cases of codimension $3$, namely, when all the three coefficients of (1.1) vanish simultaneously at some point; we call it a degenerate singular point. Degenerate singular points of differential equations of the form (1.1) were considered in [5]. In particular, a topological classification of (generic) degenerate singular points was obtained there. However, as pointed out in [5] (also see [21]), even the formal classification of degenerate singular points has functional moduli. At one time a curious phenomenon was observed, for instance, in the study of singular points of differential equations. Namely, for generic singular points (of planar vector fields) analytic classification is simple and coincides with formal classification (see [14]). In a case of codimension $1$ (for example, when there are resonances in the Siegel domain) the formal classification is still quite simple (and has a few numerical moduli) but analytic classification is no longer equivalent to formal and has functional moduli (see [22]–[24]). However, in more degenerate cases even formal classification has functional moduli; paradoxically, in these cases analytic classification coincides with formal again (as a rule). This phenomenon (when formal equivalence ensures analytic equivalence) is called rigidity. Rigidity was originally discovered for finitely generated groups of germs of one-dimensional diffeomorphisms (the CMR-theorem in Ilyashenko’s terminology: see [25]); also topological rigidity (when topological equivalence ensures analytic equivalence) was established for this problem (see [26]). Subsequently, rigidity was also proved for some classes of degenerate singular points of planar vector fields: see [25] and [27]–[31]. As there is some parallelism in results for singular points of planar vector field and singular points of binary differential equations, it was conjectured that there can also be rigidity for degenerate singular points of binary differential equations. Our aim in this paper is to verify this thesis. We consider the complex version of the classification problem for degenerate singular point of binary differential equations. Let $\mathcal{BD}$ (from ‘binary degenerate’) be the class of all equations of the form (1.2), with coefficients $a$, $b$ and $c$ that are germs of holomorphic functions in $(\mathbb{C}^2,0)$ and satisfy the degeneracy conditions Definition 1. We say that two germs in $\mathcal{BD}$ (with coefficients $(a,b,c)$ and $(A,B,C)$, respectively) are analytically equivalent if there exists a germ of a biholomorphic map $H\colon (\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$, $H(0,0)=(0,0)$, $H\colon (x,y)\mapsto(X,Y)$, and a germ of a holomorphic function $K\colon (\mathbb{C}^2,0)\to\mathbb{C}$, $K(0,0)\ne0$, such that the quadratic form can be obtained from the quadratic form by the substitution $X=X(x,y)$, $Y=Y(x,y)$ and multiplication by $K$: Then we say that the germ of the biholomorphic map $H$ conjugates these two germs in $\mathcal{BD}$. In a similar way we define the formal equivalence of two germs in $\mathcal{BD}$ (with $H$ and $K$ being formal power series and (1.4) treated as an equality of formal power series). The main result of our paper is the following rigidity theorem. Theorem 1 (on rigidity). For generic germs in $\mathcal{BD}$ formal equivalence ensures analytic equivalence. Remark 1. As in [27], the proof of the rigidity theorem is based on the CMR-theorem and on the unsolvability of the corresponding monodromy group (which holds generically). Remark 2. The numerous genericity conditions in the rigidity theorem are mainly imposed on the linear parts of germs in $\mathcal{BD}$ and will explicitly be written out in the proof of the theorem. However, there is also an implicit condition imposed on nonlinear terms of the germ (namely, on its 8-jet), which ensures that the corresponding monodromy group is unsolvable. We verify that all these conditions are generic in the respective sections. In § 10 we present an example of a germ satisfying all these conditions. Remark 3. The rigidity theorem also holds in the real case. Moreover, the complex case is even more important in a certain sense: the real version of the rigidity theorem can be derived from the complex one. However, to do this we must relax slightly the generic conditions used in Theorem 1; see Remark 9. This paper is organized as follows. In §§ 2–7 we introduce the main structures we need to work with the germ in question from the class $\mathcal{BD}$. In § 8 we present the main formulae for these structures. In § 9 we present an example of a germ in $\mathcal{BD}$ with unsolvable monodromy group; on this basis, in § 10 we prove that the condition ‘the monodromy group of the germ in $\mathcal{BD}$ is unsolvable’ is generic. In § 12 we discuss the question of equivalence for germs in $\mathcal{BD}$ and some structures linked to this question; in particular, we present a theorem stating that two germs in $\mathcal{BD}$ that have the same linear part and analytically equivalent monodromy groups are analytically equivalent (Theorem 2). This is an analogue of the results in [19] and [11] on ‘sharpening of equivalence’. Theorem 2 is in fact central to our paper; we prove it in § 12. However, as shown in § 13, the formal equivalence of generic germs in $\mathcal{BD}$ implies the formal equivalence of their monodromy groups. Taking account of the results in § 10 and the CMR-theorem, the main result of our paper (Theorem 1) follows directly from Theorem 2; see § 14. We dedicate this work to the memory of Arlen Mikhailovich Il’in, a remarkable researcher and personality, our colleague and teacher for life. § 2. The surface and submanifold of an equationWe identify a germ (1.1) in the class $\mathcal{BD}$ with the corresponding implicit equation (1.2) and call it a binary degenerated equation (a BDE for short). Definition 2. By the surface of a binary differential equation $F(x,y,p)=a(x,y)p^2+2b(x,y)p+c(x,y)=0$ we mean the set $M_F\subset (\mathbb {C}^2,0)\times \mathbb {C}$ described by this equation: Lemma 1. For a generic germ in $\mathcal{BD}$ we can select a representative $F$ so that the following hold. Proof. 0. ‘Closing’ the surface of the equation. On the variety $\mathbb{C}^2\times\mathbb{C}P^1$ we can use the atlas with two charts $(x,y,p)\in \mathbb{C}^2\times \mathbb{C}$ and $(x,y,q)\in \mathbb{C}^2\times \mathbb{C}$ and with transition function $q=p^{-1}$. In the first chart $M_F$ has the equation $\{F(x,y,p)=0\}$, and in the second the equation where $q\in\mathbb{C}\setminus\{0\}$. Adding to $M_F$ the ‘lacking’ points (ones on the curve $\gamma_{\infty}$ defined by the equations $a(x,y)=0$ and $q=0$ in the second chart) we obtain the extended surface $\mathcal{M}_F=M_F\cup\gamma_{\infty}$. Thus, $\mathcal{M}_F$ has equation (1.2) in the first chart and equation (2.1) (where now $q\in\mathbb{C}$) in the second.
1. The smoothness of $\mathcal{M}_F$. Consider the partial derivatives $F'_x(0,0,p)=:P_1(p)$ and $F'_y(0,0,p)=:P_2(p)$. By the implicit function theorem $M_F$ is a smooth surface in a neighbourhood of a point $(0,0,p_0)\in \mathbb{P}$ if at least one of these partial derivatives is distinct from zero in a neighbourhood of $p_0\in \mathbb{C}$. Let $R=R[P_1,P_2]$ be the resultant of the quadratic polynomials $P_1$ and $P_2$. We will assume that Then $P_1$ and $P_2$ have no common zeros, so for each $p_0\in \mathbb{C}$ at least one quantity, $P_1(p_0)$ or $P_2(p_0)$, is nonzero, which ensures the smoothness of $M_F$ in a neighbourhood of $(0,0,p_0)$. Moreover, it also follows from (2.2) that $\max\{\deg P_1,\deg P_2\}=2$. However, then $a'_x(0,0)\ne0$ or $a'_y(0,0)\ne0$, so that the implicit function theorem ensures that the surface defined by (2.1) is smooth in a neighbourhood of the point $x=y=0$, $q=0$. Thus, $\mathcal{M}_F$ is a smooth surface in a neighbourhood of each point on $\mathbb{P}$. Reducing the domains of definition of $a$, $b$ and $c$ if necessary we obtain the required smoothness of the whole of $\mathcal{M}_F$.2. The second assertion is a direct consequence of (1.3). 3. The third assertion of the lemma can be verified by straightforward calculations, but we will obtain it free of charge: see Corollary 1 and Remark 7. The proof is complete. On the submanifold $\mathcal{M}_F$ we use the following charts. (1) The chart $\tau_1=(x,p)$: it works for points on $M_F$ at which $F'_y\ne0$; then $y=f_1(x,p)$ can be expressed from (1.2) by the implicit function theorem. (2) The chart $\tau_2=(y,p)$: its domain consists of the points on $M_F$ at which $F'_x\ne0$; then $x=f_2(y,p)$ can be expressed from (1.2) by the implicit function theorem. (3) (For $\deg P_2=2$.) The chart ${\tau_3=(x,q)}$: it works in a small neighbourhood of the point $(x= 0,y= 0,q= 0)$, in which we can uniquely express $y=f_3(x,q)$ from (2.1). (4) (For $\deg P_1=2$.) The chart $\widetilde{\tau}_3=(y,q)$: it works in a small neighbourhood of the point $(x\,{=}\,0,y\,{=}\,0,q\,{=}\,0)$, in which we can uniquely express $x=f_4(y,q)$ from (2.1). Reducing the domains of definition of the functions $a$, $b$ and $c$ if necessary we ensure that the functions $f_i$ are well defined and the corresponding transition functions are holomorphic. Remark 4. In fact, Lemma 1 defines the germ $(\mathcal{M}_F,\mathbb{P})$ of the submanifold $\mathcal{M}_F$ that corresponds to the given germ in $\mathcal{BD}$ and is defined in the natural way. The same holds for the charts defined above. Then all arguments related to the ‘reduction’ of domain of definition are consistent. For less cumbersome verbal expressions we abuse the terminology slightly and drop the word ‘germ’ here and occasionally in what follows. Definition 3. We call the submanifold $(\mathcal{M}_F,\mathbb{P})$ constructed in Lemma 1 from germs in $\mathcal{BD}$ the submanifold of the equation in question; we call the above charts on $(\mathcal{M}_F,\mathbb{P})$ natural charts on $(\mathcal{M}_F,\mathbb{P})$. § 3. Linearizing germs from $\mathcal{BD}$Definition 4. The linear part (or linearization) of a germ $F\in\mathcal{BD}$ is the germ in $\mathcal{BD}$ of the form where the quadratic polynomials $P_1$ and $P_2$ are defined in terms of $F$ by $P_1(p)=F'_x(0,0,p)$ and $P_2(p)=F'_y(0,0,p)$. As we saw in § 2, the generic assumptions made about $F\in\mathcal{BD}$ in Lemma 1 are in fact assumptions about its linear part; they are as follows. Condition G1. The resultant $R=R[P_1,P_2]$ of the polynomials $P_1$ and $P_2$ defining the linear part of $F\in\mathcal{BD}$ given by (1.2) and (1.3) is distinct from zero. Remark 5. Direct calculations (also see [5], p. 257) yield the following: if $P_1(p)=a_1p^2+2b_1p+c_1$ and $P_2(p)=a_2p^2+2b_2p+c_2$, then Throughout what follows condition G1 is assumed to hold. § 4. Foliation on the submanifold of the equationFor $F\in\mathcal{BD}$ the contact hyperplanes $dy-pdx=0$ cut a direction field $\xi_F$ on the surface $M_F$ of the equation. Let $\mathcal{F}_F$ be the foliation of $M_F$ by the phase curves of this direction field. Lemma 2. A generic binary differential equation (1.2) satisfies the following conditions. 1. The foliation $(\mathcal{F}_F,M_F)$ extends to a holomorphic foliation (with singularities) $(\mathfrak{F}_F,\mathcal{M}_F)$ on the submanifold of the equation. 2. The foliation $(\mathfrak{F}_F,\mathcal{M}_F)$ has precisely three singular points: these singular points lie on the sphere $\mathbb{P}$. 3. The foliation $(\mathfrak{F}_F,\mathcal{M}_F)$ is nondicritical: the punctured sphere $\mathbb{P}_F$ obtained from $\mathbb{P}$ by the removal of the singular points of $\mathfrak{F}_F$ is a leaf of $\mathfrak{F}_F$. Proof. 1. It is easy to verify (see [14]) that for the implicit equation (1.2) the vector field $V_F=-F'_p\,{\partial}/{\partial x}-pF'_p\,{\partial}/{\partial y}+(F'_x+pF'_y)\,{\partial}/{\partial p}$ is tangent to the surface of the equation $M_F=\{F=0\}$ at its points and lies in the kernel of the contact from $\omega=dy-p\,dx$. Hence the direction field $\xi_F$ is spanned (where it is well defined) by (the restriction to $M_F$ of) this vector field. Thus, $\xi_F$ is a holomorphic direction field (in the union of the domains of the charts $\tau_1$ and $\tau_2$). Note now that the field $V_F$ (expressed in the variables $(x,y,p)$ on $\mathbb{C}^2\times\mathbb{C}P^1$) is different from $\widehat{V}_F=-q\widehat{F}'_q\,{\partial}/{\partial x}-\widehat{F}'_q\,{\partial}/{\partial y}+(q\widehat{F}'_x+\widehat{F}'_y)\,{\partial}/{\partial q}$ on $M_F$ (here $\widehat{F}=a+2bq+cq^2$, as in Lemma 1) by a single factor of $-q^{-1}$. Since $\widehat{V}_F$ is holomorphic in a neighbourhood of $(x=y=0,p=\infty)\in\mathbb{C}^2\times\mathbb{C}P^1$ and the direction field $\xi_F$ is spanned by $\widehat{V}_F$ at points in $M_F$, this direction field extends to a holomorphic direction field defined on the whole of $\mathcal{M}_F$. This yields immediately the first assertion of the lemma.
2. We find singular points of $(\mathfrak{F}_F,\mathcal{M}_F)$ on the sphere $\mathbb{P}$ (all other singular points can be eliminated by a standardized reduction of the domains of definition of the coefficients of germs in $\mathcal{BD}$). Clearly, they are singular points of $V_F$ (plus maybe the point $x=y=0$, $p=\infty$ if it is singular for $\widehat{V}_F$). It follows from (1.3) that $F'_p=0$ at the points in $\mathbb{P}$. Hence $(0,0,p_0)\in\mathbb{P}$ is a singular point of $\mathfrak{F}_F$ if and only if $F'_x+pF'_y=0$ at this point. In the notation of § 3 this means that $P(p)=P_1(p)+pP_2(p)$ vanishes at $p_0$: $P(p_0)=0$. For a germ in $\mathcal{BD}$ with generic linear part the polynomial $P$ is cubic (which holds for $\deg P_2=2$) and has precisely three different zeros (provided that its discriminant is distinct from zero). It remains to observe that if $\deg P_2=2$, then ($x=y=0$, $q=0$) is a smooth point of $\widehat{V}_F$. This proves the second assertion of Lemma 2. 3. The third assertion is a direct consequence of (1.3). The proof is complete. Remark 6. Thus, Lemma 2 defines the (germ of the) holomorphic foliation $(\mathcal{F}_F,\mathcal{M}_F)$ on the submanifold of the germ $F\in\mathcal{BD}$. We call it the foliation induced by the germ $F$. § 5. Singular points of the foliation $(\mathfrak{F}_F,\mathcal{M}_F)$As shown in Lemma 2, points $(0,0,p_0)$ are singular for the foliation $\mathfrak{F}_F$ if the polynomial $P(p)=F'_x+pF'_y|_{x=y=0}=P_1+pP_2$ vanishes at $p_0$. We impose some extra generic conditions (used in the above proof of Lemma 2) on the linear part of (1.2). Condition G3. The discriminant $\Delta_P$ of the polynomial $P=F'_x+pF'_y|_{x=y=0}$ is distinct from zero. Note that the first part of Condition G2 yields $\deg P_2=2$, so $\deg P=3$. Let $p_j$, $j=1,2,3$, be zeros of $P$ (all of which are different by Condition G3). By the second part of Condition G2 we have $P(0)=P_1(0)\ne0$, so that $p_j\ne0$, $j=1,2,3$. Now, if $P(p_j)=0$, then $P_1(p_j)=-p_jP_2(p_j)$. Since $p_j\ne0$ and the polynomials $P_1$ and $P_2$ have no common zeros, it follows that $P_1(p_j)\ne0$ and $P_2(p_j)\ne0$. Thus we have proved the following result. Lemma 3. Under Conditions G1–G3 of genericity the singular points $(0,0,p_j)$, $j=1,2,3$, lie in the domains of definition of both the charts $\tau_1$ and $\tau_2$. In addition, We find the characteristic exponents $\lambda_j$ (that is, ratios of eigenvalues of the linearization of the field producing the foliation) at the singular points $(0,0,p_j)$ of the foliation $\mathfrak{F}_F$. In the chart $\tau_1=(x,p)$, $y=f_1(x,p)$ the field $V_1=V_F|_{\mathcal{M}_F}$ has the form
$$
\begin{equation*}
V_1(x,p)=-F'_p\,\frac{\partial}{\partial x}+(F'_x+pF'_y)\,\frac{\partial}{\partial p}\bigg|_{y=f_1(x,p)}.
\end{equation*}
\notag
$$
If $P_1=F'_x|_{x=y=0}$ and $P_2=F'_y|_{x=y=0}$, then from the equation $F(x,y,p)=xP_1+yP_2+\dotsb=0$ we find that $y=f_1(x,p)=-x{P_1(p)}/{P_2(p)}+O(x^2)$ as $x\to0$. However, then
$$
\begin{equation*}
\begin{aligned} \, F'_p|_{y=f_1(x,p)} &=xP'_1(p)+f_1(x,p)P'_2+O(x^2) \\ &=x\frac{P'_1(p)P_2(p)-P'_2(p)P_1(p)}{P_2(p)}+O(x^2), \qquad x \to0. \end{aligned}
\end{equation*}
\notag
$$
We also have $F'_x+pF'_y|_{y=f_1(x,p)}=P(p)+O(x)$ as $x\!\to\!0$, where $P(p)\!=\!P_1(p)+ pP_2(p)$. Hence the linear part of the field $V_1$ at $x=0$, $p=p_j$ has a lower triangular matrix with diagonal entries
$$
\begin{equation*}
\frac{-P'_1(p_j)P_2(p_j)+P'_2(p_j)P_1(p_j)}{P_2(p_j)}
\end{equation*}
\notag
$$
and $P'(p_j)$, where $P'(p_j)\ne0$ by Condition G3. Therefore,
$$
\begin{equation}
\lambda_j=\frac{P'_2(p_j)P_1(p_j)-P'_1(p_j)P_2(p_j)}{P_2(p_j)P'(p_j)}.
\end{equation}
\tag{5.2}
$$
On the other hand $0=P(p_j)=P_1(p_j)+p_jP_2(p_j)$, so that $P_1(p_j)=-p_jP_2(p_j)$. Moreover, $P'(p_j)=P'_1(p_j)+p_jP'_2(p_j)+P_2(p_j)$. Using these relations, from (5.2) we obtain However, $p=p_j$ is a simple zero of the rational function ${P_2}/{P}$. That is, we have proved the following result. Corollary 1. The sum of characteristic exponents of singular points of $\mathfrak{F}_F$ is equal to $-2$: Proof. Because
$$
\begin{equation*}
\frac{P_2(p)}{P(p)}=\frac{P_2(p)}{pP_2(p)+P_1(p)}=\frac{1}{p}+O\biggl(\frac{1}{p^2}\biggr), \qquad p\to \infty,
\end{equation*}
\notag
$$
it follows that $\operatorname{res}_{\infty}{P_2(p)}/{P(p)}=-1$. Now the equality follows from Lemma 4 by the residue theorem.
The corollary is proved. Remark 7. Note that $\lambda_j$ is precisely equal to the Camacho–Sad index (see [32]) of the singular point $(0,0,p_j)$ of the foliation $\mathfrak{F}_F$ with respect to the leaf $\mathbb{P}_F$. By the Camacho–Sad theorem [32], Corollary 1 shows that the self-intersection index of the sphere $\mathbb{P}$ is $-2$. Remark 8. Note that the characteristic exponents are determined by the linear part of (1.2); see (5.3). In what follows we look at binary differential equations (1.2) satisfying an additional constraint. It follows from Remark 8 that this is a restriction on the linear part of the binary equation, which holds for generic equations (1.2) by Lemma 4. Remark 9. Note that Condition G4 ensures that all singular points of $\mathfrak{F}_F$ are linearizable and nonresonant, and this is what we will need below. Hence we can relax Condition G4 (by replacing it by generic conditions on singular points imposed in [11]). While Condition G4 never holds in the real-valued case, the relaxed condition is fulfilled for generic real binary differential equations. § 6. The involutions $I^F$, $I_F$ and $i_F$On the surface $M_F$ of the equation we consider the involution $I^F$ interchanging the roots of the quadratic equation $F=0$. Lemma 5. For a generic equation (1.2) the following hold. 1. The involution $I^F$ extends to a biholomorphic involutive map $I_F\colon (\mathcal{M}_F,\mathbb{P})\to(\mathcal{M}_F,\mathbb{P})$, $I^F\circ I_F=\mathrm{id}$, with the following properties. 2. The sphere $\mathbb{P}$ is $I_F$-invariant ($I_F(\mathbb{P})=\mathbb{P}$). 3. Let $i_F=I_F|_{\mathbb{P}}$. Then $i_F$ is a Möbius transformation of $\mathbb{P}$ and ${i_F\circ i_F=\mathrm{id}_{\mathbb{P}}}$. The fixed points $a^{\pm}$ of $i_F$ are distinct, lie in the common domain of the charts $\tau_1$ and $\widetilde{\tau}_3$ and differ from the singular points of $\mathfrak{F}_F$:
$$
\begin{equation*}
a^{+}\ne a^-, \qquad P_1(a^{\pm})\ne0\quad\textit{and}\quad P(a^{\pm})\ne0.
\end{equation*}
\notag
$$
4. The fixed point set $\operatorname{Fix}(I_F)$ of $I_F$ consists of two smooth curves $l^{\pm}_{F}$ that intersect $\mathbb{P}$ transversally at the fixed points $a^{\pm}$ of $i_F$:
$$
\begin{equation*}
\operatorname{Fix}(I_F)=l^{+}_F\cup l^{-}_{F}, \qquad l^{\pm}_{F}\pitchfork \mathbb P\quad\textit{and}\quad \operatorname{Fix}(I_F)\cap\mathbb{P}=\operatorname{Fix}(i_F).
\end{equation*}
\notag
$$
5. At each point in $\operatorname{Fix}(I_F)$ the eigenvector corresponding to the eigenvalue $-1$ of the linear part of $I_F$ is tangent to the leaf of $\mathfrak{F}_F$ passing through this point. Proof. 1. By Vieta’s theorem, in the chart $(x,p)$, $y=f_1(x,p)$, the involution $I^F$ acts by the formula $I^F(x,p)=(x,\widetilde{p})$, where
$$
\begin{equation}
\widetilde{p}=-\frac{2b(x,y)}{a(x,y)}-p\Big|_{y=f_1(x,p)}.
\end{equation}
\tag{6.1}
$$
Let $a(x,y)=a_1x+a_2y+\dotsb$, $b(x,y)=b_1x+b_2y+\dotsb$ and $c(x,y)=c_1x+c_2y+\dotsb$, so that (using the notation from the previous sections) $P_1(p)=a_1p^2+2b_1p+c_1$, $P_2(p)=a_2p^2+2b_2p+c_2$, $P(p)=P_1(p)+pP_2(p)$ and $y=f_1(x,p)=-x{P_1(p)}/{P_2(p)}+O(x^2)$ as $x\to0$. Then from (6.1) we obtain
$$
\begin{equation}
\widetilde{p}=\widetilde{p}(x,p) =-p-2\frac{b_1-b_2{P_1(p)}/{P_2(p)}}{a_1-a_2{P_1(p)}/{P_2(p)}}+O(x)=i(p)+O(x), \qquad x\to0,
\end{equation}
\tag{6.2}
$$
where
$$
\begin{equation}
i(p)=-\frac{p\Delta_2+\Delta_1}{p\Delta_3+\Delta_2},
\end{equation}
\tag{6.3}
$$
and $\Delta_1$, $\Delta_2$ and $\Delta_3$ are the determinants of the matrices obtained from $\begin{pmatrix} a_1 & 2b_1 & c_1\\ a_2 & 2b_2 & c_2 \end{pmatrix}$ by deleting the first, second and third columns, respectively. It follows from (6.2) that the limit exists (at any rate, for $p$ at which $p\Delta_3+\Delta_2\ne0$). Repeating similar calculations in the chart $\tau_3=(x,q)$, $y=f_3(x,q)$ for $I^F$ and setting $I^F\colon (x,q)\mapsto(x,\widetilde{q})$ we obtain
$$
\begin{equation}
\widetilde{q}=\widetilde{q}(x,q)=-\frac{2b(x,y)}{c(x,y)}-q\Big|_{y=f_3(x,q)}=\widetilde{i}(q)+O(x),\qquad x\to0,
\end{equation}
\tag{6.4}
$$
where
$$
\begin{equation}
\widetilde{i}(q)=-\frac{q\Delta_2+\Delta_3}{\Delta_2+q\Delta_1}.
\end{equation}
\tag{6.5}
$$
It follows from (6.4) that the limit exists (at any rate, for $q$ such that $\Delta_2+\Delta_1q\ne0$). It remains to observe that the exceptional value $q_{*}=-{\Delta_2}/{\Delta_1}$ on $\mathbb{P}$ corresponds to the point with coordinates $(x=0,p=\widetilde{p}_*=-{\Delta_1}/{\Delta_2})$ in the chart $(x,p)$. But $\widetilde{p}_*$ is distinct from the other exceptional value $p_*=-{\Delta_2}/{\Delta_3}$ (for otherwise $\Delta_1\Delta_3=\Delta_2^2$, which contradicts Condition G1 in view of (3.2)). Thus, $I^F$ extends to all points on $\mathbb{P}$ by continuity. We impose additionally the generic condition $\Delta_2\ne0$. Then $\Delta_2+\Delta_1q\ne0$ for small $q$, and we see from (6.4) and (6.5) that the limit $\lim _{q\to0} \widetilde{q}(x,q)=-{\Delta_3}/{\Delta_2}$ exists. This means that $\widetilde{q}(x,q)$ can be extended to the line $q=0$ by continuity. Thus, $I^F$ extends to a map $I_F$ defined in a neighbourhood of the whole sphere $\mathbb{P}$. By theorems on exceptional sets (see [33]) the extended map is holomorphic in $(\mathcal{M}_F,\mathbb{P})$. It is an involution by continuity since $I^F$ is an involution. This completes the proof of the first assertion of the lemma.
2, 3. The second assertion of the lemma follows from the explicit formulae (6.2)–(6.5) representing $I_F$ in different charts. Hence it is immediate that $i_F=I_F\big|_{\mathbb{P}}$ is an involution. Moreover, (6.3) and (6.5) give explicit expressions for $i_F$ (in the charts $p$ and $q=p^{-1}$ on $\mathbb{P}$). $3'$. Fixed points of $i_F$. As follows from the expression (6.3) for $i_F$ (in the chart $p$), the fixed points of $i_F$ are precisely the roots of the equation We assume additionally that $\Delta_1\ne0$ and $\Delta_3\ne0$; then (6.6) is a quadratic equation with nontrivial discriminant (because $\Delta_2^2\ne \Delta_1\Delta_3$, as mentioned above). Hence (6.6) has two (distinct) roots. Let these be $a^{+}$ and $a^{-}$; then $a^{\pm}\ne0$, $a^{\pm}\ne\infty$,
$$
\begin{equation}
\mathcal{D}(a^{\pm})=0 \quad\text{and}\quad \mathcal{D}'(a^{\pm})\ne0.
\end{equation}
\tag{6.7}
$$
Note that $\mathcal{D}=P'_1P_2-P'_2P_1$. Then it follows from (5.2) and Condition G4 that the fixed points $a^{\pm}$ of $i_F$ are distinct from the singular points of $\mathfrak{F}_F$. We assume additionally that $P_1$ and $P_2$ have no multiple zeros. Then $P'_k(p)=0\Rightarrow P'_k(p)\ne0$, $k=1,2$, and in view of Condition G1 we also have $P_1(a^{\pm})\ne0$ and $P_2(a^{\pm})\ne0$, so both fixed points of $i_F$ occur in the domain of definition of the chart $\tau_1$ (and of $\widetilde{\tau}_3$).
4. Fixed point of $I_F$. In view of the above, fixed points of $I_F$ lie in the domain of the chart $(x,p)$, $y=f_1(x,p)$. It follows from the first part of the proof that the fixed point set $\operatorname{Fix}(I_F)$ consists of all pairs $(x,p)$ such that where $\widetilde{p}(x,p)$ is from (6.2). In particular, $\psi(0,a^{\pm})=0$. But $\psi'_p(0,a^{\pm})={i'(a_{\pm})-1=-2}$ (because $a_{\pm}$ are fixed points of the one-dimensional involution $i$). Hence by the implicit function theorem (6.8) is solvable with respect to $p$ in a neighbourhood of the points $x=0,p=a^{\pm}$. This proves the fourth assertion of Lemma 5.5. Note that (in the chart $\tau_1$) $I_F$ preserves the ‘straight lines’ $\{x=\mathrm{const}\}$, and its restriction to such a line is a one-dimensional involution. Hence the eigenvector with eigenvalue $-1$ of the linearization of $I_F$ at the fixed point is ‘vertical’ (that is, collinear with ${\partial}/{\partial p}$). However, fixed points of $I_F$ are multiple zeros of $F$, so $F'_p=0$ at these points. This means that the vector field $V_F$ generating the foliation $\mathfrak{F}_V$ is ‘vertical’ at these points. This proves the last assertion of the lemma. Lemma 5 is proved. Remark 10. In Lemma 5 we imposed additionally the following generic conditions. Condition G5. The following minors are nondegenerate: $\Delta_1\ne0$, $\Delta_2\ne0$ and $\Delta_3\ne0$. A further generic condition (which is, in fact, also imposed on the linear part of a binary differential equation) will be convenient to use in what follows. § 7. Correspondence maps and monodromy transformations. Monodromy group $G_F$The foliation $(\mathfrak{F}_F,\mathcal{M}_F)$ corresponding to a generic bilinear equation is nondicritical (Lemma 2), so the correspondence map and monodromy transformation are well defined (for instance, see [34]). Recall the relevant definitions. Let $a \in \mathbb{P}_F$ be a nonsingular point of the foliation $\mathfrak{F}_F$. We can rectify $\mathfrak{F}_F$ in a neighbourhood of $a$: there exists a neighbourhood $U=U(a)\subset \mathcal{M}_F$ and a diffeomorphism $H$ of $U$ onto the bidisc $\mathbb{D}^2=\{(z,w)\in \mathbb{C}^2\colon |z|<1,\,|w|<1\}$, $H(a)=(0,0)$, that takes leaves of $\mathfrak{F}_F$ (more precisely, of its restriction to $U$) to leaves of the standard foliation $\{w=\mathrm{const}\}$ on $\mathbb{D}^2$. Let $T_1$ and $T_2$, $T_{1,2}\subset U$, be transversals to $\mathbb{P}$ at nonsingular points $a_1=T_1\cap\mathbb{P}$ and $a_2=T_2\cap\mathbb{P}$, respectively, and let $\widetilde{T}_k=H(T_k)$, $k=1,2$. Then the curve $\widetilde{T}_k$ is transversal to the disc $\mathbb{D}=\{w=0\}\cap \mathbb{D}^2$ at $b_k=H(a_k)$. Hence for all sufficiently small $c\in\mathbb{C}$ the leaf $\{w=c\}$ intersects each $\widetilde{T}_k$ at a unique point $W_k(c)$. Then the map $\Delta_{1,2}\colon H^{-1}(W_1(c))\mapsto H^{-1}(W_2(c))$ is well defined (at any rate, in a small neighbourhood of $a_1$ on the transversal $T_1$). The resulting map $\Delta_{1,2}\colon (T_1,a_1)\to(T_2,a_2)$ is the correspondence map (for the transversals $T_1$ and $T_2$). It is a local holomorphic diffeomorphism if $T_1$ and $T_2$ are holomorphic curves (which is just the case in our setting). The above construction certainly work if the base points of the transversals coincide: $a_1=a_2$. In the general case we can proceed as follows: we connect $a_1$ with $a_2$ by a curve $\gamma\subset \mathbb{P}_F$ and consider intermediate transversals $S_1,\dots,S_N$ with base points $A_m=S_m\cap\mathbb{P}_F$ on $\gamma$ such that $S_1=T_1$, $S_N=T_2$ and the correspondence maps $\Delta_{m,m+1}\colon (S_m,A_m)\to(S_{m+1},A_{m+1})$ are well defined for all $m=1,\dots,N-1$. Then by the correspondence map $\Delta_{\gamma}^{T_1T_2}\colon (T_1,a_1)\to (T_2,a_2)$ we mean the composition of the $\Delta_{m,m+1}$:
$$
\begin{equation*}
\Delta_{\gamma}^{T_1T_2}=\Delta_{N-1,N}\circ\dots\circ\Delta_{1,2}.
\end{equation*}
\notag
$$
In the case when $\gamma$ is a closed curve on $\mathbb{P}_F$ and $T_1=T_2=:T$ we call the correspondence map $\Delta^{TT}_{\gamma}$ the monodromy transformation (of the foliation $\mathfrak{F}_F$ corresponding to the transversal $T$ and the curve $\gamma$). When the transversals we use are clear, we drop the superscripts in the notation for correspondence maps (and monodromy transformations). In the above definition we have some freedom (in the choice of the intermediate transversals and rectifying diffeomorphisms). Remarkably (for instance, see [34]) the germs $\mathbf{\Delta}$ of the resulting maps $\Delta$ are well defined (independent of the above choice). Note the most important properties of correspondence maps ([34]): (1) $\gamma_1$ is homotopic to $\gamma_2$ on $\mathbb{P}_F$ (written $\gamma_1\sim\gamma_2$) $\Rightarrow \mathbf{\Delta}_{\gamma_1}=\mathbf{\Delta}_{\gamma_2}$; (2) $\mathbf{\Delta}_{\gamma^{-1}}=(\mathbf{\Delta}_{\gamma})^{-1}$; (3) $\mathbf{\Delta}_{\gamma_1\gamma_2}=\mathbf{\Delta}_{\gamma_2}\circ\mathbf{\Delta}_{\gamma_1}$. It follows from the first property that, in particular (for each transversal $T$ with nonsingular base point $a=T\cap\mathbb{P}$), there is a well-defined map $J_a$ that assigns to each element $\gamma$ of the fundamental group $\pi_1(\mathbb{P}_F,a)$ a germ $\mathbf\Delta_{\gamma}$ in the local diffeomorphism group $\operatorname{Diff}(T,a)$: The other two properties show that the map $J_a$ is an (anti)representation of $\pi_1(\mathbb{P}_F,a)$ into $\operatorname{Diff}(T,a)$. Taking a parameter $z$ on $T$ such that $z(a)=0$, we can identify $(T,a)$ with $(\mathbb{C},0)$ (and $\operatorname{Diff}(T,a)$ with $\operatorname{Diff}=\operatorname{Diff}(\mathbb{C},0)$).Definition 5. We call the above map $J_a\colon \pi_1(\mathbb{P}_F,a)\to \operatorname{Diff}$ the monodromy (anti)representation, and we call its image $G_F:=\{\mathbf{\Delta}_{\gamma}\colon \gamma\in \pi_1(\mathbb{P}_F,a)\}$ the monodromy group of the foliation $\mathfrak{F}_F$. Remark 11. There is still considerable freedom in the definition of the monodromy group (namely, the choice of the transversal and the parameter on it). For example, let $\widetilde{T}$ be another transversal to $\mathbb{P}_F$ (with the same base point $a$), and let $\widetilde{z}$ be a parameter on $\widetilde{T}$, $\widetilde{z}(a)=0$ (in particular, it is possible that $T$ coincides with $\widetilde{T}$, just their parametrizations are different). Let $G_F$ and $\widetilde{G}_F$ be the monodromy groups corresponding to the transversals $T$ and $\widetilde{T}$. Let $\textbf{h}\colon (\mathbb{C},0)\to(\mathbb{C},0)$ be the correspondence map $(T,a)\to(\widetilde{T},a)$ expressed in terms of the variables $z$ and $\widetilde{z}$. Then the groups $G_F$ and $\widetilde{G}_F$ are conjugate: if $\gamma\in\pi_1(\mathbb{P}_F,a)$, $\mathbf{\Delta}_{\gamma}\in G_F$ and ${\widetilde{\mathbf{\Delta}}_{\gamma}\in\widetilde{G}_F}$, then A similar relation holds in the general case. Namely, let $(T,a)$ and $(\widetilde{T},\widetilde{a})$ be two transversals to $\mathbb{P}_F$, $z$ and $\widetilde{z}$ be parameters on these transversals, and $G_F$ and $\widetilde{G}_F$ be the corresponding monodromy groups. Let $\gamma_0\subset\mathbb{P}_F$ be a curve connecting the base points $a$ and $\widetilde{a}$ and $\mathbf{h}\colon (\mathbb{C},0)\!\to\!(\mathbb{C},0)$ be the correspondence map from ${(T,a)\!\approx\!(\mathbb{C},0)}$ to $(\widetilde{T},\widetilde{a})\!\approx\!(\mathbb{C},0)$ corresponding to this curve. Let $j_0\colon \gamma\!\mapsto\!\gamma_0^{-1}\gamma\gamma_0$ be the corresponding natural isomorphism between the fundamental groups $\pi_1(\mathbb{P}_F,a)$ and $\pi_1(\mathbb{P}_F,\widetilde{a})$. Then it immediately follows from properties (1)–(3) of correspondence maps that Thus, assuming that elements of the monodromy group are numbered by elements of the relevant fundamental group, the above relation means that elements of the monodromy groups found for different transversals are conjugate (after a suitable renumbering) by a local diffeomorphism (which is the same for all elements). Remark 12. The punctured sphere $\mathbb{P}_F$ of a generic binary equation (1.2) is obtained from $\mathbb{P}$ be the removal of three singular points (see Lemma 2). Hence $\pi_1(\mathbb{P}_F,a)$ is a free group on two generators. As such generators we can take two curves $\gamma_1$ and $\gamma_2$, each of which encircles (precisely once) the (precisely one) singular point $p_1$ or $p_2$ of the foliation $\mathfrak{F}_F$. Then $G_F$ is a finitely generated group with generators $\mathbf{\Delta}_1:=\mathbf{\Delta}_{\gamma_1}$ and $\mathbf{\Delta}_2:=\mathbf{\Delta}_{\gamma_2}$: $G_F=\langle \mathbf{\Delta}_1,\mathbf{\Delta}_2\rangle $. Definition 6. We say that two finitely generated groups $G=\langle \mathbf{\Delta}_1,\mathbf{\Delta}_2\rangle $ and $\widetilde{G}=\langle\mathbf{\widetilde{\Delta}_1},\mathbf{\widetilde{\Delta}}_2\rangle$, $G,\widetilde{G}\subset \operatorname{Diff}(\mathbb{C},0)$, are analytically equivalent ($G\stackrel{\text{an}}{\sim} \widetilde{G}$) if there exists a germ $\mathbf{h}\in \operatorname{Diff}(\mathbb{C},0)$ conjugating their generators: The above argument shows that the monodromy group $G_F$ of the binary equation $F$ is not uniquely defined, but its class of analytic equivalence $[G_F]$ is well defined. § 8. Formulae for computationsIn this section we deduce formulae required for an explicit computation of the structures defined above for generic binary differential equations (1.2). 8.1. Singular points and their characteristic exponentsAs shown in § 4, the singular points of the foliations $\mathfrak{F}_F$ and the characteristic exponents at these points are determined by the linear part of the equation. To find singular points we must solve cubic equations, after which the characteristic exponents can be found explicitly; see § 4. 8.2. The standard chart $\tau_1=(x,p)$, $y=f_1(x,p)$For the standard chart $\tau_1$ the function $y=f_1(x,p)$ is determined from (1.2) by means of the implicit function theorem:
$$
\begin{equation*}
y=f_1(x,p)\quad\Longleftrightarrow\quad F:=a(x,y)p^2+2b(x,y)p+c(x,y)=0.
\end{equation*}
\notag
$$
In the domain of $\tau_1$ (that is, where $F'_y(0,0,p)\ne0$) the function $f_1$ is analytic and has an expansion Remark 13. Note that the ‘linear part’ $f_{11}(p)$ is determined by the linear part of the binary equation. Moreover, for each $n\geqslant1$ the $n$-jet of $f$ (the $n$th partial sum of the series in (8.1)) is uniquely determined from the $n$-jet of equation (1.2). In addition, the coefficients $f_{1n}(p)$ are rational functions of $p$ with coefficients found by arithmetic operations from the components of the (at most) $n$-jets of the functions $a,b$ and $c$. 8.3. The involutions $I_F$ and $i_F$In § 6, for the involutions $I_F$ and $i_F$ (which in the chart $\tau_1$ coincide with $I^F(x,p)=(x,\widetilde{p})$ and $i$, respectively) we obtained formulae (6.1)–(6.3). The functions $\widetilde{p}$ is analytic and has an expansion Note that:$\bullet$ the free term $i_0(p)$ of the expansion (8.2) is precisely equal to the involution $i_F$; it is determined by the linear part of (1.2); $\bullet$ for each $n\geqslant1$ the $n$-jet of $\widetilde{p}$ (that is, the $n$th partial sum of the series in (8.2)) is uniquely determined by the $n$-jet of equation (1.2). An observation similar to Remark 13 also holds here. 8.4. The monodromy transformationUnder Conditions G1–G4 foliation $\mathfrak{F}_F$ has precisely three singular points $p_1,p_2$ and $p_3$, $p_j\ne0$ (see § 5), and all of these lie in the domain of the chart $\tau_1=(x,p)$, $y=f_1(x,p)$ (defined for $P_2(p):=F'_y(0,0,p)\ne0$). Hence leaves of the foliation $\mathfrak{F}_F$ are (in the chart $\tau_1$) phase curves of the restriction $\widetilde{V}_F$ of the field $V_F$ to $M_F$ (see § 4). However, these phase curves are integral curves of the differential equation
$$
\begin{equation}
\frac{dx}{dp}=-\frac{F'_p}{F'_x+pF'_y}\Big|_{y=f_1(x,p)}.
\end{equation}
\tag{8.3}
$$
In the notation of § 5 the right-hand side of (8.3) has the form
$$
\begin{equation*}
\begin{aligned} \, Q(x,p) &:=-\frac{F'_p}{F'_x+pF'_y}\Big|_{y=f_1(x,p)} \\ &=-2\frac{a(x,y)p+b(x,y)}{(a'_x+pa'_y)p^2+2p(b'_x+b'_y)+c'_x+c'_yp}\Big|_{y=f_1(x,q)} \\ &=-2\frac{p\cdot a(x,f_1(x,p))+b(x,f_1(x,p))}{P(p)+O(x)}. \end{aligned}
\end{equation*}
\notag
$$
Expanding $Q$ in a power series in $x$, we obtain so that $Q_1$ is indeed defined by the linear part of the binary equation. Taking into account part $3'$ of the proof of Lemma 5 and (6.6), we obtain Accordingly, for $k\geqslant2$ we see that $Q_k(p)$ is a rational function with coefficients determined by the $k$-jet of the original binary equation, and with poles (at the $p_j$, $j=1,2,3$, and maybe also at zeros of $P_2(p)$). We construct the monodromy group $G_F$ for the transversal $T=\{p=0\}$; as a parameter on the transversal we take $x$. Let $x=x(p,x_0)$ be the solution of (8.3) with initial condition $(x_0,p_0)$: let $x(0,x_0)=x_0$. By the theorem on the differentiable dependence of solutions on the initial data the function $x(p,x_0)$ is analytic and has a power expansion in $x_0$: From the condition $x(0,x_0)=x_0$, for the $X_k$ we obtain the initial conditions Substituting (8.4) and (8.6) into (8.3) and equating the coefficients of like powers of $x_0$, for the $X_k$ we obtain an infinite system of equations
$$
\begin{equation}
\begin{aligned} \, \dot{X_0}&=0, \\ \dot{X_1} &=Q_1X_1, \\ \dot{X_2} &=Q_1X_2+Q_2X_1^2, \\ &\dots\dots\dots\dots\dots \\ \dot{X}_k &=Q_1X_k+\psi_k, \\ &\dots\dots\dots\dots\dots, \end{aligned}
\end{equation}
\tag{8.8}
$$
where $\psi_k$ is a polynomial of $X_1,\dots,X_{k-1}$ with coefficients $Q_2,\dots,Q_k\colon \psi_k=\psi_k[X_1,\dots,X_{k-1},Q_2,\dots,Q_k]$. For $k=0$, from (8.8), taking (8.7) into account we obtain $X_0\equiv0$ (as it must be for a nondicritical foliation). For $k=1$, from (8.8), taking (8.7) into account we obtain For $k=2$ the corresponding solution of (8.8) is linear inhomogeneous. Solving it by the method of variation of a constant, taking (8.7) into account we obtain In a similar way, for $k\geqslant3$ we have where the function $R_k(s)$ is obtained from the polynomial $\psi_k$ by substituting in the functions $X_j=X_j(s)$, $1\leqslant j\leqslant k-1$, which we have already calculated. Now let $\gamma\subset \mathbb{P}_F$ be a closed curve on the sphere $\mathbb{P}_F$ that connects the points $0$ and $p$. Then formulae (8.9)–(8.11) define the analytic continuation of the solution $x(p,x_0)$ along the curve $\gamma$ (where integration in (8.10) and (8.11) is performed along $\gamma$). Hence if $\Delta_{\gamma}\in G_F$ is the monodromy transformation corresponding to the path $\gamma$, then the coefficients $\Delta_{\gamma,k}$ of the expansion $\Delta_{\gamma}(x_0)=\sum_{k=1}^{\infty} \Delta_{\gamma,k}x_0^k$ are defined, in accordance with (8.9)–(8.11), by the formulae
$$
\begin{equation*}
\begin{gathered} \, \Delta_{\gamma,1}=\exp\biggl(\int_{\gamma} Q_1(s)\,ds\biggr), \\ \Delta_{\gamma,2}=\int_{\gamma} Q_2(s)X_1(s)\,ds \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\Delta_{\gamma,k}=\int_\gamma X_1^{-1}R_k(s)\,ds, \qquad k\geqslant3.
\end{equation*}
\notag
$$
It follows from the explicit formulae for coefficients of the monodromy transformation that the following result holds. Lemma 6. The linear part of the monodromy transformation is determined by the linear part of the binary differential equation. For each $n\geqslant1$ the $n$-jet of the monodromy transformation is determined by the $n$-jet of the equation, and its coefficients are (generally speaking, multivalued) analytic functions of the coefficients of the $n$-jet of the binary equation. Remark 14. We see that the ‘linear parts’ $X_1$ of the solution $x(t,x_0)$ are determined by the linear part of the binary equation alone. Hence the (local) first integral $J(x,p)=x_0$ of the solution (expressing $x_0$ in terms of $(x,p)$ on the basis of (8.6)) has the same property: its linear part $J(x,p)=xA_0(p)+\dotsb$ is determined by the linear part of the linear equation: $A_0(p)=(X_1(p))^{-1}$. 8.5. The symmetric foliationBelow we will need the foliation $\mathfrak{F}^*_{F}\,{=}\,I_F(\mathfrak{F}_F)$, ‘symmetric’ to $\mathfrak{F}_F$. In the chart $\tau_1$ it is generated by the vector field $V^{*}_{F}=(I'_F\cdot V_F)\circ I_F$ (the image of $V_F$ under the involution $I_F$). After fairly cumbersome calculations (in the chart $\tau_1=(x,p)$, $y=f_1(x,p)$), for the field $V^*_F$ we obtain
$$
\begin{equation}
\begin{aligned} \, \notag V^*_F(x,p) &=\biggl(x\frac{\mathcal{D}(p)}{P_2(p)}+O(x^2)\biggr) \frac{\partial}{\partial x} \\ &\qquad +\biggl(\frac{\mathcal{D}(p)P_2(p)-P(p)(\Delta_3p+\Delta_2)}{(\Delta_3p+\Delta_2)}+O(x)\biggr) \frac{\partial}{\partial p}, \qquad x\to0. \end{aligned}
\end{equation}
\tag{8.12}
$$
Remark 15. Phase curves of the vector field $V_F^*=(V^*_1,V^*_2)$ are equal to integral curves of the nonautonomous differential equation ${dx}/{dp}={V^*_1}/{V_2^*}=:Q^*$. It follows from (8.12) that $Q^*(x,p)=xQ_1^*(p)+O(x^2)$ as $x\to0$, where § 9. Example of a BDE with unsolvable monodromy groupConsider a binary differential equation of the following form: We work in the chart $\tau_1=(x,p)$. From (9.1) we obtain $y=f_1(x,p)= -{(p^2+2p+2)}x-\alpha p^2(p^2+2p+2)^2x^2-2\alpha p^2(p^2+2p+2)^3x^3+O(x^4)$. Substituting $y=f_1(x,p)$ into $V_1(x,p)=-F'_p\,{\partial}/{\partial x}+(F'_x+pF'_y)\,{\partial}/{\partial p}$ we obtain the system
$$
\begin{equation*}
\begin{cases} \dot{x}=-2(p+1)x-2\alpha p(p^2+2p+2)^2x^2-4\alpha^2p^3(p^2+2p+2)^3x^3+O(x^4), \\ \dot{p}=(p+1)(p+2)-2\alpha p^3(p^2+p+2)x-2\alpha^2 p^5(p^2+2p+2)^2x^2 \\ \qquad -4\alpha^2 p^5(p^2+2p+2)^3x^3+O(x^4). \end{cases}
\end{equation*}
\notag
$$
Hence where
$$
\begin{equation*}
\begin{aligned} \, Q_1&=-\frac{2}{p+2}, \\ Q_2&=-\frac{2\alpha p(p^2+2p+2)^2}{(p+1)(p+2)}-\frac{4\alpha p^3(p^2+2p+2)}{(p+1)(p+2)^2} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, Q_3&=-\frac{4\alpha^2 p^3(p^2+p+2)^3}{(p+1)(p+2)}-\frac{4\alpha^2p^5(p^2+2p+2)^2}{(p+1)(p+2)^2} \\ &\qquad -\frac{4\alpha^2 p^4(p^2+2p+2)^3}{(p+1)^2(p+2)^2} -\frac{8\alpha^2 p^6(p^2+2p+2)^2}{(p+1)^2(p+2)^3}. \end{aligned}
\end{equation*}
\notag
$$
In accordance with § 8, we represent the solution $x=x(p,x_0)$ of (9.2) with initial data $x(0,x_0)=x_0$ in the form
$$
\begin{equation*}
x=x(p,x_0)=\sum_{k=1}^{\infty} X_k(p)x_0^k=X_1x_0+X_2x_0^2+X_3x_0^3+\dotsb.
\end{equation*}
\notag
$$
Then $X_1(0)=1$, $X_2(0)=0$, $X_3(0)=0$, $\dots$ and
$$
\begin{equation*}
\begin{cases} \dot{X}_1=Q_1X_1, \\ \dot{X}_2=Q_1X_2+Q_2X_1^2, \\ \dot{X}_3=Q_1X_3+2Q_2X_1X_2+Q_3X_1^3. \end{cases}
\end{equation*}
\notag
$$
From the first equation, as $X_1(0)=1$, we obtain We solve the second equation of the system by the variation of a constant. The solution of the corresponding homogeneous differential equation has the form Then Integrating, we obtain
$$
\begin{equation*}
C_1(p)=-8\alpha\int_0^p \frac{q(q+2)(q^2+2q+2)^2+2q^3(q^2+2q+2)}{(q+1)(q+2)^4}\,dq.
\end{equation*}
\notag
$$
Thus, the solution has the form
$$
\begin{equation}
X_2(p)=-\frac{32\alpha}{(p+2)^2}\int_{0}^{p}f(q)\,dq,
\end{equation}
\tag{9.4}
$$
where $f(q)=({q(q+2)(q^2+2q+2)^2+2q^3(q^2+2q+2)})/({(q+1)(q+2)^4})$. We also solve the third equation by the variation of a constant. The solution of the corresponding homogeneous differential equation has the form Then
$$
\begin{equation*}
\dot{C}_2(p)=2Q_2(p)X_2(p)+Q_3(p)X_1^2(p), \qquad C_2(0)=0.
\end{equation*}
\notag
$$
Hence, taking (9.3) into account we obtain Thus, the solution of the third equation has the form
$$
\begin{equation}
X_3(p)=X_1(p)C_1^2(p)+X_1(p)\int_{0}^{p}Q_3(q)X_1^2(q)\,dq,
\end{equation}
\tag{9.5}
$$
where
$$
\begin{equation*}
\begin{gathered} \, X_1(p)=\frac{4}{(p+2)^2}, \qquad C_1(p)=-8\alpha\int_{0}^{p} f(q)\,dq, \\ f(q)=\frac{q(q+2)(q^2+2q+2)^2+2q^3(q^2+2q+2)}{(q+1)(q+2)^4}, \\ \begin{split} \int_{0}^{p}Q_3(q)X_1^2(q)\,dq &=-64\alpha^2\int_{0}^{p}f_1(q)\,dq -64\alpha^2\int_{0}^{p}f_2(q)\,dq \\ &\qquad -64\alpha^2\int_{0}^{p}f_3(q)\,dq -128\alpha^2\int_{0}^{p}f_4(q)\,dq, \end{split} \\ f_1(q)=\frac{q^3(q^2+2q+2)^3}{(q+1)(q+2)^5}, \qquad f_2(q)=\frac{q^5(q^2+2q+2)^2}{(q+1)(q+2)^6}, \\ f_3(q)=\frac{q^4(q^2+2q+2)^3}{(q+1)^2(q+2)^6}, \qquad f_4(q)=\frac{q^6(q^2+2q+2)^2}{(q+1)^2(q+2)^7}. \end{gathered}
\end{equation*}
\notag
$$
We calculate the monodromy transformation for the transversal $T_0=\{p=0\}$ using the $x$-coordinate as a parameter on the transversal. Let $\gamma\in\pi_1(\mathbb{P}_F,0)$ (so that $\gamma$ is a closed curve on the plane $\mathbb{C}\setminus\{-1,-2\}$ beginning at $p=0$ and $\Delta_{\gamma}\colon T_0\to T_0$ is the corresponding monodromy transformation). Then
$$
\begin{equation}
\Delta_{\gamma}(x_0)=\sum_{k=1}^{\infty} x_0^k X_{k\gamma},
\end{equation}
\tag{9.6}
$$
where $X_{k\gamma}$ is the result of the analytic continuation of $X_k$ along $\gamma$. Let $\gamma_1$ be a curve encircling only the point $p=-1$ (in the positive direction) and $\gamma_2$ be a curve encircling only the point $p=-2$ (in the positive direction). Calculating the integrals in (9.4) and (9.5) by use of residues we obtain
$$
\begin{equation*}
\begin{gathered} \, X_{1\gamma_1}=1,\qquad X_{2\gamma_1}=48\pi \alpha i,\qquad X_{3\gamma_1}=-2304\pi^2\alpha^2+4864\pi\alpha^2i; \\ X_{1\gamma_2}=1,\qquad X_{2\gamma_2}=0\quad\text{and}\quad X_{3\gamma_2}=-9728\pi\alpha^2i. \end{gathered}
\end{equation*}
\notag
$$
Thus, for the generators $\Delta_1=\Delta_{\gamma_1}$ and $\Delta_2=\Delta_{\gamma_2}$ we obtain
$$
\begin{equation}
\Delta_1(x)=x+48\pi\alpha i x^2+(-2304\pi^2\alpha^2+4864\pi\alpha^2 i)x^3+\dotsb
\end{equation}
\tag{9.7}
$$
and Note that $\Delta_1$ and $\Delta_2$ are parabolic germs of distinct orders (see [34]): $\Delta_1$ has order $2$, and $\Delta_2$ has order $3$. Hence (see [34], Proposition 6.11) $\Delta_1$ and $\Delta_2$ do not commute. Thus we have established the following result. Lemma 7. The monodromy group of equation (9.1) is non-Abelian. However, for germs with identity linear part, if two germs do not commute, then they generate an unsolvable group (see [25], Proposition 2.1). Corollary 2. The monodromy group of equation (9.1) is unsolvable. § 10. Unsolvability of the monodromy groups of generic binary differential equations (1.2)Recall that a group $G$ is said to be solvable if the sequence of commutants $\{G_k\}$, $G_0=G$, $G_{k+1}=[G_k,G_k]=\langle [f,g]\colon f,g\in G_k\rangle $ terminates: for some $k$ the group $G_k$ is trivial (consists of the identity element). For groups of germs of one-dimensional holomorphic diffeomorphisms there is a simple criterion of solvability. Lemma 8 (solvability criterion; see [25] or [34]). A group $G\subset \operatorname{Diff}(\mathbb{C},0)$ is solvable if and only if two arbitrary elements of $G$ with identity linear parts commute ($\Leftrightarrow$ its commutant $G_1$ is Abelian $\Leftrightarrow$ $G_2$ is trivial: $G_2=\{\mathrm{id}\}$). Definition 7 (see [34]). For a germ $f(z)=z+cz^n+\dotsb$ tangent to the identity map ($n\geqslant2$) such that $c\ne0$ we call $n$ the order (of tangency to the identity map). It is easy to show that two germs with distinct tangency orders $m$ and $n$ never commute (their commutator has order $m+n-1$: see [34]; Proposition 6.11). Example 1. The germs $g_1=\Delta_1$ and $g_2=\Delta_2$ from § 9 have orders 2 and 3, respectively (see (9.7) and (9.8)), so that the commutator $g_3=[g_1,g_2]$ has order $4$. Hence the germ $g_4=[g_2,g_3]$ has order 5 and $g_5=[g_3,g_4]$ has order 8: Lemma 9. For a generic binary differential equation (1.2) its monodromy group is unsolvable. Proof. Let $\mathcal{BD}_3$ be the class of germs in $\mathcal{BD}$ that satisfy Conditions G1–G3 of genericity. Let $F_0$ be the differential equation from § 9; note that $F_0$ does not belong to $\mathcal{BD}_3$ (Condition G2 fails), but lies in its closure. Let $F$ be an equation in the class $\mathcal{BD}_3$. Since $\mathcal{BD}_3$ is a connected set (the subsets forbidden by Conditions G1–G3 have complex codimension 1 — and real codimension 2,–in the space of linear parts) $F$ and $F_0$ can be connected by a curve $\Gamma=\{F_t\}_{t\in[0,1]}$, $F_1=F$, lying in $\mathcal{BD}_3$ (apart from the endpoint $F_0$). Let $P_1^t(p)={\partial F_t}/{\partial x}(0,0,p)$, $P_2^t={\partial F_t}/{\partial y}(0,0,t)$ and $P^t(p)=P_1^t(p)+pP_2^t(p)$. By Condition G3 the zeros $p_1(t)$, $p_2(t)$ and $p_3(t)$ of the equation $P^t(p)=0$ are distinct for all $t\in(0,t]$, and therefore they depend continuously on $t$. We number them so that $p_1(0)=1$ and $p_2(0)=2$ (that is, in accordance with the numbering in the example in § 9).
Note that the point $x=0$, $y=0$, $p=0$ is nonsingular for all $F_t$, $t\in[0,1]$ (as follows from Conditions G1–G3). Let $\gamma_1^0$ and $\gamma_2^0$ be generators of the fundamental group $\pi_1(\mathbb{P}_{F_0},0)$ in the example in § 9 (that is, $\gamma_j^0$ is the boundary of a domain in $\mathbb{P}$ which contains the point $p_j(0)$, but does not contain other singular points $p_k(0)$, $k\ne j$, or zeros of $P_2^0$, $j=1,2$). For $t\in(0,1]$ we select in a similar way generators $\gamma_1^t$ and $\gamma_2^t$ of the fundamental groups $\pi_1(\mathbb{P}_{F_t},0)$ (so that $\gamma_j^{t}$ is a boundary of a domain in $\mathbb{P}$ which contains $p_j(t)$ but does not contain other singular points $p_k(t)$, $k\ne j$, or zeros of $P_2^t$; we can take these curves to depend continuously on $t\in[0,1]$). Let $\Delta_j^t\colon (\mathbb{C},0)\to(\mathbb{C},0)$, $j=1,2$, be the monodromy transformation for the equation $F_t$ calculated for the transversal $T_0=\{p=0\}$ ($x$ is a parameter on $T_0$) in accordance with § 8 (and then $\Delta_1^0$ and $\Delta_2^0$ are just the monodromy transformations from the example in § 9). Set $g_3^t=[\Delta_1^t,\Delta_2^t]$, $g_4^t=[\Delta_2^t,g_3^t]$ and $g_5^t=[g_3^t,g_4^t]$, so that $g^0_5$ is precisely equal to (10.1). Let $c_k^t$ be a coefficient of the Taylor expansion $g_5^t(x)=\sum_{k=1}^{\infty} c_k^t x^k$; it follows from (10.1) that By Lemma 6 all coefficients of the 8-jet of the monodromy transformation are uniquely determined by the 8-jet of the original binary equation and are (multivalued) analytic functions of the coefficients of this jet. Hence the coefficient $c_8^t$ is an analytic function of the coefficients of the 8-jet of $F_t$. Therefore, $c_8^1$ is an analytic function of the coefficients of the equation $F_1=F$ (obtained from $c^0_8$ by ‘analytic continuation along $\Gamma$’). On the other hand, the condition of vanishing of a nontrivial analytic function (by (10.2)) distinguishes a subset of codimension 1 (in the space of coefficients of the 8-jet of equation (1.2)). Hence, for a generic binary equation $F\in\mathcal{BDE}_3$ the corresponding coefficient $c_8^1=c_8^1(F)$ is not zero. This implies that the corresponding germs $g_3^1$ and $g_4^1$ do not commute. Both germs have the identity linear part (since they are commutators) and belong to $G_F$. Hence the monodromy group $G_F$ is unsolvable by the criterion of solvability (Lemma 8). The proof is complete. Remark 16. We can see from the proof of Lemma 9 that the generic condition ensuring that the monodromy group of a germ in $\mathcal{BD}$ is unsolvable deals with the eighth jet of the germ. Example 2 (of germs in $\mathcal{BD}$ satisfying all the conditions of genericity). Let $F_0(p)=(x+\alpha y^2)p^2+2xp+2x+y$ be the germ from § 9, $F_{\varepsilon}(p) =F_0(p) + {\varepsilon}p^2y$. Then for all sufficiently small and not purely real $\varepsilon$ the germ $F_{\varepsilon}(p)$ satisfies all the conditions of genericity imposed on germs in $\mathcal{BD}$. § 11. Equivalence of binary differential equations and related structures11.1. Analytic equivalence of binary differential equations and associated foliationsThe following result is obvious. Lemma 10. 1. A local holomorphic diffeomorphism $H$ conjugating two binary differential equations $F$ and $\widetilde{F}$ satisfying Conditions G1–G3 extends to a holomorphic diffeomorphism $\widehat{H}\colon (\mathcal{M}_F,\mathbb{P})\to(\mathcal{M}_{\widetilde{F}},\mathbb{P})$ of the submanifolds of these binary equations. Moreover, the following hold. 2. $\widehat{H}$ takes the sphere $\mathbb{P}$ to itself, and its restriction $\widehat{h}: =\widehat{H}\big|_{\mathbb{P}}$ is a linear fractional isomorphism between $\mathbb{P}_F$ and $\mathbb{P}_{\widetilde{F}}$. 3. The homomorphism $\widehat{H}$ conjugates the foliations $\mathfrak{F}_F$ and $\mathfrak{F}_{\widetilde{F}}$ (that is, it takes leaves of one foliation to leaves of the other). 4. The homomorphism $\widehat{H}$ conjugates the involutions $I_F$ and $I_{\widetilde{F}}$: We call $\widehat{H}$ a pullback of the corresponding diffeomorphism $H$. In a certain sense this is indeed a pullback of $H$ to the (almost) two-sheeted covers $\mathcal{M}_F$ and $\mathcal{M}_{\widetilde{F}}$ with respect to the natural projections $\pi_F\colon (\mathcal{M}_F,\mathbb{P})\to(\mathbb{C}^2,0)$ and $\pi_{\widetilde{F}}\colon (\mathcal{M}_{\widetilde{F}},\mathbb{P})\to(\mathbb{C}^2,0)$ ‘forgetting’ the $p$-coordinate: The result below is also quite simple. Lemma 11. Let $\widehat{H}\colon (\mathcal{M}_F,\mathbb{P})\to(\mathcal{M}_{\widetilde{F}},\mathbb{P})$ be a diffeomorphism conjugating the foliations $\mathfrak{F}_F$ and $\mathfrak{F}_{\widetilde{F}}$ (in the sense of assertions 2 and 3 of Lemma 10), and conjugating the involutions $I_F$ and $I_{\widetilde{F}}$ (that is, assertion 4 of Lemma 10 also holds). Then the binary equations $F$ and $\widetilde{F}$ are analytically equivalent, and moreover, $\widehat{H}$ is a pullback of a local diffeomorphism $N$ conjugating these equations. Proof. Since $\widehat{H}$ conjugates the involutions $I_F$ and $I_{\widetilde{F}}$, $\widehat{H}$ takes orbits of the former to ones of the latter. However, the full preimages of points in a punctured neighbourhood of the origin $(\mathbb{C}^2,0)\setminus(0,0)=:(\mathbb{C}_*^2,0)$ under the projections $\pi_{F}$ and $\pi_{\widetilde{F}}$ are precisely orbits of the corresponding involutions. Therefore, the map $H\colon (\mathbb{C}_*^2,0)\to(\mathbb{C}_*^2,0)$ for which (11.1) holds on $(\mathbb{C}_*^2,0)$ is well defined. We define $H$ at zero by setting $H(0)=0$; for this extension of $H$ equality (11.1) holds in a full neighbourhood of the origin. It is easy to see that this map is continuous in $(\mathbb{C}^2,0)$ and holomorphic (at any rate, away from the projection $\pi_F(\operatorname{Fix}I_F)$ of the mirror of $I_F$). Hence by results on removable singularities $H$ is holomorphic in $(\mathbb{C}^2,0)$. The fact that $H$ is invertible follows from the invertibility of $\widehat{H}$. Since solutions of a binary differential equation are the projections of leaves of the corresponding foliation onto the submanifold of this equation, the condition that $\widehat{H}$ conjugates $\mathfrak{F}_F$ and $\mathfrak{F}_{\widetilde{F}}$ implies that $H$ takes solutions of the binary equation $F$ to solutions of $\widetilde{F}$. Hence $F\stackrel{\text{an}}{\sim} \widetilde{F}$, $H$ is a local holomorphic diffeomorphism conjugating these equations, and $\widehat{H}$ is a pullback of $H$.
The proof is complete. 11.2. Analytic equivalence of foliations and their monodromy groupsThe monodromy group $G_F$ of a binary differential equation $F$ is in fact determined by the corresponding induced foliation $\mathfrak{F}_F$. We will use the concept of ‘conjugacy’ for such foliations (see Lemma 11) and the definition of equivalence from § 7 for finitely generated groups of germs. The following result was in fact established above (see the end of § 7). The converse is also true (under certain additional conditions). Lemma 13. Let two binary differential equations $F$ and $\widetilde{F}$ have the same linear part, which satisfies Conditions G1–G4. If their monodromy groups are analytically equivalent, then the induced foliations are conjugate: Proof. Below we will prove a much stronger result, so here we limit ourself to a brief sketch of the proof.
It follows from Conditions G1–G3 that the foliation $\mathfrak{F}_F$ has precisely three singular points $A_j=(0,0,p_j)\in\mathbb{P}$, $j=1,2,3$. By Condition G4 there is a separatrix $S_j$ through each of these points, which is transversal to the sphere $\mathbb{P}\colon S_j\pitchfork\mathbb{P}$ (the sphere $\mathbb{P}$ itself is the second separatrix, common to all singular points). We construct an analytic field of transversals $\{T_a\}_{a\in\mathbb{P}}$ to $\mathbb{P}$ that contains the separatrices $T_{A_j}=S_j$. The dicritical foliation $J_F=\{T_a\}$ is transversal to $\mathfrak{F}_F$ outside the separatrix set $S_F=S_1\cup S_2 \cup S_3$:
$$
\begin{equation*}
J_F \pitchfork \mathfrak{F}_F\quad\text{on}\quad\mathcal{M}_F\setminus S_F=:\mathring{\mathcal{M}}_F.
\end{equation*}
\notag
$$
Hence the pair of foliations $(J_F,\mathfrak{F}_F)$ produces in the domains $\mathring{\mathcal{M}}_F$ the ‘coordinates’ $(a,z)$, $a\in\mathbb{P}$, $z\in(\mathbb{C},0)$, defined as follows. Let $\Pi_F\colon \mathcal{M}_F\to\mathbb{P}$ be the projection along leaves of the foliation $J_F\colon \Pi_F(A)=a\Leftrightarrow A\in T_a$. For a point $A\in \mathring{\mathcal{M}}_F$ consider a path $\gamma(A)$ (on $\mathbb{P}$) that connects the point $a=\Pi(A)$ with $0$, and let $\widehat{\gamma}_A$ be the lift of this path from the base $\mathbb{P}$ to the leaf $L_A$ of $\mathfrak{F}_F$ passing through $A$. The endpoint $B$ of $\widehat{\gamma}_A$ lies on the transversal $T_0$. Let $z\colon (T_0,0)\to(\mathbb{C},0)$ be a parameter on $T_0$. Then the pair $\tau(A)=(a=\Pi(A),z=z(B))$ is equal to the required coordinates of $A \in \mathring{\mathcal{M}}_F$.
In a similar way, using the appropriate auxiliary dicritical foliation $\widetilde{\mathfrak{J}}_{\widetilde{F}}$ and the projection $\widetilde{\Pi}_{\widetilde{F}}$ we construct a coordinate system $\widetilde{\tau}=(\widetilde{a},\widetilde{z})$ on $\mathring{\mathcal{M}}_{\widetilde{F}}$. Now we can define a diffeomorphism $\widehat{H}$ conjugating $\mathfrak{F}_F$ with $\mathfrak{F}_{\widetilde{F}}$ in a natural way:
$$
\begin{equation*}
\forall\, A\in \mathring{\mathcal{M}}_{F} \quad \forall \,\widetilde{A}\in \mathring{\mathcal{M}}_{\widetilde{F}}\colon \quad \widehat{H}(A)=\widetilde{A} \quad\Longleftrightarrow \quad \tau(A)=\widetilde{\tau}(\widetilde{A}).
\end{equation*}
\notag
$$
The (possible) ambiguity in the definition of $\widehat{H}$ is eliminated by the equivalence of the monodromy groups $G_F$ and $G_{\widetilde{F}}$; $\widehat{H}$ extends from $\mathring{\mathcal{M}}_{F}$ to the whole of $\mathcal{M}_F$ because the linear parts of $F$ and $\widetilde{F}$ coincide (and therefore the singular points of the foliations $\mathfrak{F}_F$ and $\mathfrak{F}_{\widetilde{F}}$, as well as their characteristic exponents, coincide; see Remark 8).
The proof is complete. Remark 17. Note that it is very important in the above proof that a dicritical foliation transversal to $\mathfrak{F}_F$ exists outside the separatrix set $S_F$. 11.3. The analytic equivalence of binary equations and their monodromy groupsThe following result is a direct consequence of Lemmas 10 and 12. Lemma 14. The analytic equivalence of two generic binary differential equations implies that their monodromy groups are conjugate. Unfortunately, a similar combination of Lemmas 11 and 13 does not allow us to establish the result converse to Lemma 14: a diffeomorphism $\widehat{H}$ in Lemma 13 that conjugates the foliation $\mathfrak{F}_F$ with $\mathfrak{F}_{\widetilde{F}}$ does not necessarily conjugate the involutions corresponding to these equations, so we cannot use Lemma 11. Nevertheless, this result is valid, although for its proof we require a much more refined construction than the one in Lemma 13. Theorem 2. If two binary differential equations with equal linear parts satisfying Conditions G1–G7 have analytically equivalent monodromy groups, then these binary equations are analytically equivalent. This result is actually the most interesting part of our paper. We prove it in § 12. § 12. The proof of Theorem 212.1. A lemma on five transversalsIt is known that five smooth curves in $(\mathbb{C}^2,0)$ that intersect transversally at the origin can be straightened simultaneously by a local diffeomorphism (for instance, see [35]). Hence by Grauert’s theorem (see [36]) we can simultaneously ‘straighten’ any four transversals to the submanifold $\mathbb{P}\approx\mathbb{C}P^1$ of the complex manifold $\mathcal{M}$, provided that $\mathbb{P}\cdot\mathbb{P}=-1$. We show that when the self-intersection index of $\mathbb{P}$ is $-2$, we can do the same even for five transversals. Since for the submanifold $(\mathcal{M}_F,\mathbb{P})$ of a generic binary equation $F$ we have $\mathbb{P}\cdot\mathbb{P}=-2$ (see Lemma 1) and, according to Grauert [36], all neighbourhoods $\mathcal{M}$ of a sphere $\mathbb{P}\approx\mathbb{C}P^1$ such that $\mathbb{P}\cdot\mathbb{P}=-2$ are pairwise equivalent, we can state and prove the corresponding result precisely in this particular case. Lemma 15. Let $F$ be a generic binary differential equation and $(\mathcal{M}_F,\mathbb{P})$ be its submanifold from Lemma 1. Let $(T_j,a_j)$, $j=1,2,3,4,5$, be transversals to the sphere $\mathbb{P}$ at five different points: $a_j \ne a_k$ for $k\ne j$. Then there exists a ‘local’ holomorphic diffeomorphism $\mathcal{H}\colon (\mathcal{M}_F,\mathbb{P})\to(\mathcal{M}_F,\mathbb{P})$ equal to the identity on the sphere $\mathbb{P}$, $\mathcal{H}|_{\mathbb{P}}=\mathrm{id}$, and taking the $T_j$ to the ‘straight lines’ $T_j^0=\{p=a_j\}$, ${j=1,\dots,5}$. Proof. (a) First we consider the simplest binary equation $F_0=0$, where The assumptions of Lemma 1 are fulfilled for $F_0$, and on the manifold $\mathcal{M}_{F_0}$ we can do with just two charts (which are standard charts in the terminology of § 2), $\tau_1=(x,p)$ and $\widetilde{\tau}_3=(y,q)$. The corresponding transition function $\Phi$ can be found explicitly: $(y,q)=\Phi(x,p)\Leftrightarrow q=p^{-1}$, $y=xp^2$.
Let $a_5=\infty$ and $T_5=\{p=\infty\}$, and let the remaining points $a_j$, $j=1,2,3,4$, lie in the domain of definition of the chart $\tau_1=(x,p)$, $y=f_1(x,p)=xp^2$. Let $p=\gamma_j(x)$ be the equation of the transversal $T_j$; then $\gamma_j(x)=a_j+\alpha_j(x)+O(x)$ as $x\to0$, where $\alpha_j(0)=0$. Consider the Lagrange interpolation polynomial
$$
\begin{equation*}
L(x,p)=\sum _{j=1}^4 \alpha_j(x) \prod _{k\ne j} \frac{p-a_k}{a_j-a_k}.
\end{equation*}
\notag
$$
Then $L(x,a_j)=\alpha_j(x)$, $j=1,2,3,4$, so that the map takes each straight line $T^0_j=\{p=a_j\}$ to the respective transversal $T_j,H(T^{0}_j)=T_j$, $j=1,2,3,4$. Moreover, since $\alpha_j(x)=O(x)$ as $x\to0$, it follows that $L(x,p)=O(x)$ as $x\to0$, and therefore $H(0,p)=(0,p)$ for all $p\in\mathbb{C}$. The map (12.1) has the form $\widetilde{H}\colon (y,q)\mapsto(\widetilde{y},\widetilde{q})$ in the chart $(y=xp^2,q=p^{-1})$, where $\widetilde{y}=y(1+q\widetilde{L})^{-2}$, $\widetilde{q}=q(1+q\widetilde{L})^{-1}$ and $\widetilde{L}=L(yq^2,q^{-1})$. However, $L(x,p)$ is a polynomial of degree at most three (in $p$), which vanishes identically for $x=0$. Hence $q\widetilde{L}=qL(yq^2,q^{-1})$ is also a polynomial in $q$ (of degree at most 3), which vanishes identically for $y=0$. Consequently, the map $\widetilde{H}$ continues analytically to the line $\{q=0\}$ (and preserves it: $\widetilde{q} = 0$ for $q=0$). Thus we have a well-defined local diffeomorphism $\widehat{\mathcal{H}}$ (equal to $H$ in the chart $(x,p)$ and to $\widetilde{H}$ in the chart $(y,q)$) which takes the ‘straight’ transversals $T^{0}_j=\{p=a_j\}$ to the given transversals $T_j$. The inverse diffeomorphism $\mathcal{H}=\widehat{\mathcal{H}}^{-1}$ is the required ‘straightening’.
(b) The general case reduces to the above special case. In fact, we can first straighten $T_5$ (by mapping it to $T^{0}_5=\{p=a_5\}$); then take it to infinity by an appropriate linear fractional’ isomorphism $\mathbf{h}\colon (x,p)\to(x,h(p))$; then straighten, in accordance with (a), the images of the other four transversals, and then take everything back by the inverse maps $\mathbf{h}^{-1}$. It remains to observe that, according to Grauert, all neighbourhoods of a sphere $\mathbb{C}P^1$ with self-intersection index $-2$ are pairwise equivalent. Hence the result on straightening five transversals which we established for the equation $F_0$ can be extended to any other binary equations (for which the conclusion of Lemma 1 holds). The lemma is proved. In fact, rather than Lemma 15 itself, for further constructions we need its corollary stated below. Corollary 3. For a generic binary differential equation $F$ and five transversals $(T_j,a_j)\subset(\mathcal{M}_F,\mathbb{P})$ to the sphere $\mathbb{P}$ (with distinct base points $a_j$), $1\leqslant j \leqslant 5$, there exists an analytic field of transversals $T_F=\{T_a\}\big|_{a\in\mathbb{P}}$ to $\mathbb{P}$ (that is, a dicritical foliation) that contains the transversals under consideration: $T_{a_j}=T_j$. Proof. For the standard transversals $T_j=T^{0}_j=\{p=a_j\}$ the standard field ${\{p=\mathrm{const}\}}$ is the required one. In general, the required field of transversals is obtained by the straightening diffeomorphism from Lemma 15.
The proof is complete. 12.2. The scheme of construction of a conjugating diffeomorphismIn this section we give a ‘geometric’ description of a diffeomorphism $\widehat{\mathcal{H}}$ establishing the equivalence of the foliations and involutions corresponding to the binary equations in Theorem 2. Let two binary differential equations $F$ and $\widetilde{F}$ have equal linear parts satisfying Conditions G1–G6 and have equivalent monodromy groups: $G_F\stackrel{\text{an}}{\sim} G_{\widetilde{F}}$. Let $(\mathcal{M}_F,\mathbb{P})$ and $(\mathcal{M}_{\widetilde{F}},\mathbb{P})$, $\mathfrak{F}_F$ and $\mathfrak{F}_{\widetilde{F}}$, $I_F$ and $I_{\widetilde{F}}$ be the submanifolds, foliations (Lemma 2) and involutions (Lemma 5) associated with these binary equations, respectively. Since the singular points (and their characteristic exponents) are determined by the linear parts of the binary differential equations (Lemmas 3 and 4), the foliations $\mathfrak{F}_F$ and $\mathfrak{F}_{\widetilde{F}}$ have the same singular points, so we can identify the punctured spheres corresponding to these foliations: $\mathbb{P}_F=\mathbb{P}_{\widetilde{F}}=:\mathring{\mathbb{P}}$. Let $l^{\pm}_F$ and $l^{\pm}_{\widetilde{F}}$ be the mirrors of the involutions $I_F$ and $I_{\widetilde{F}}$, respectively. The curves $l^{\pm}_F$ and $l^{\pm}_{\widetilde{F}}$ are transversal to $\mathbb{P}$ and intersect it at some points $a^{\pm}_F$ and $a^{\pm}_{\widetilde{F}}$, respectively (Lemma 5). However, these points are fixed by the involutions $i_F$ and $i_{\widetilde{F}}$ from Lemma 5, and these involutions are determined by the linear parts of the binary equations (see (6.3)). Hence $i_F=i_{\widetilde{F}}=:i$ and $a^{\pm}_F=a^{\pm}_{\widetilde{F}}=:a^{\pm}$. As transversals for the calculation of the monodromy groups $G_F$ and $G_{\widetilde{F}}$, we use the mirrors $l^{+}_F$ and $l^{+}_{\widetilde{F}}$ (which have the common base point $a^{+}$). As a parameter along these transversals we use the $x$-coordinate. We identify the fundamental groups $\pi_1(\mathbb{P}_F,a^{+}_F)$ and $\pi_1(\mathbb{P}_{\widetilde{F}},a^{+}_{\widetilde{F}})$ with $\pi(\mathring{\mathbb{P}},a^{+})$ and consider the generators of $G_F=\langle \Delta_1,\Delta_2\rangle $ and $G_{\widetilde{F}}=\langle \widetilde{\Delta}_1,\widetilde{\Delta}_2\rangle $ corresponding to the same pair of generators of $\pi_1(\mathring{\mathbb{P}},a^{+})$. By assumption the groups $G_F$ and $G_{\widetilde{F}}$ are conjugate: this means that the generators $\Delta_j$ and $\widetilde{\Delta}_j$ are conjugate by some diffeomorphism $h$. Hence (after making the preliminary change $(x,y)\to(h(x),y)$ in the equation $\widetilde{F}$ and multiplying it by $h'(0)$: this does not change the linear part) we can assume that
$$
\begin{equation}
\Delta_1=\widetilde{\Delta}_1\quad\text{and} \quad \Delta_2=\widetilde{\Delta}_2.
\end{equation}
\tag{12.2}
$$
Then for each curve $\gamma\in\pi_1(\mathring{\mathbb{P}},a^{+})$ the corresponding monodromy transformations $\Delta_{\gamma}\in G_F$ and $\widetilde{\Delta}_{\gamma}\in G_{\widetilde{F}}$ also coincide:
$$
\begin{equation}
\forall\,\gamma\in\pi_1(\mathring{\mathbb{P}},a^{+}) \qquad\Delta_{\gamma}=\widetilde{\Delta}_{\gamma}.
\end{equation}
\tag{12.3}
$$
Let $p_j$ be singular points of the foliation $\mathfrak{F}_F$ (their number is precisely three by Lemma 2: $j=1,2,3$). It follows from Condition G4 that a separatrix $S^{F}_j$ of the foliation $\mathfrak{F}_F$ which is transversal to $\mathbb{P}$ passes through $p_j$. The base points $p_j$ of the transversals $(S_j^F,p_j)$ are distinct from the base points $a^{+}_{F}$ and $a^{-}_{F}$ of the transversals $l^{+}_{F}$ and $l^{-}_{F}$ (see assertion 3 of Lemma 5). Hence by Corollary 3 there exists a dicritical foliation $\mathcal{T}_F=\{T_a\}_{a\in\mathbb{P}}$ that contains all transversals $(S_j^F,p_j)$, $j=1,2,3$, and $(l^{\pm}_F,a^{\pm}_{F})$ as leaves. Let $\Pi_F\colon (\mathcal{M}_F,\mathbb{P})\to\mathbb{P}$ be the projection of $ \mathcal{M}_F$ onto $\mathbb{P}$ corresponding to this foliation (to a point $A$ on a transversal $T_a$ $\Pi_F$ assigns the base point $a$ of this transversal: $\Pi_F\colon A\mapsto a\Leftrightarrow A\in T_a$). Let $S_F=S_1^{F}\cup S_2^F \cup S_3^F$. We define a (generally speaking, multivalued) function $Z$ on $\mathring{\mathcal{M}}_F=\mathcal{M}_F\setminus S_F$ as follows. For a point $A\in \mathring{\mathcal{M}}_F$ consider its projection $a=\Pi_F(A)$ onto $\mathbb{P}$; then $a\in \mathbb{P}_F$. Consider a path $\gamma_a\subset \mathbb{P}_F$ connecting $a$ with the point $a^{+}_F$, and let $\widehat{\gamma}_A$ be its lift (with respect to $\Pi_F$) to the leaf of $\mathfrak{F}_F$ passing through $A$. Let $B$ be the endpoint of $\widehat{\gamma}_A$; it lies on $l^{+}_{F}$. Then set $Z(A)$ to be equal to $x(B)$ (recall that $x$ is the parameter on $l^{+}_F$). For the foliation $\mathfrak{F}_F$ consider the ‘symmetric’ foliation $\mathfrak{F}^{*}_F=I_F(\mathfrak{F}_F)$ (with leaves equal to the images of leaves of $\mathfrak{F}_F$ under $I_F$). It has the singular points $p^{*}_j=i_F(p_j)$, $j=1,2,3$, and the corresponding separatrices as the images $S^{F*}_j=I_F(S^F_j)$ of the separatrices $S_j^F$. Let $\mathcal{T}^{*}_F=I_F(\mathcal{T}_F)$ be the image of the dicritical foliation $\mathcal{T}_F$; then $\mathcal{T}^{*}_F$ is also dicritical, and the transversals $S_j^{F*}$, $j=1,2,3$, and mirrors $I_F(l^{\pm}_F)=e^{\pm}_F$ are leaves of it. Let $\Pi^{*}_F=\Pi_F\circ I_F$ be the projection onto the base $\mathbb{P}$ corresponding to the dicritical foliation $\mathcal{T}^{*}_F$. Precisely as we defined the function $Z$ in terms of the foliation $\mathfrak{F}_F$ and the projection $\Pi_F$, we define a function $Z^{*}$; clearly, $Z^{*}=Z\circ I_F$ (for an appropriate choice of branches of these multivalued functions). Thus, at least in some part of the domain $\mathring{\mathcal{M}}_F\cap\mathring{\mathcal{M}^{*}_F}$, where $\mathring{\mathcal{M}^*_F}=I_F(\mathring{\mathcal{M}}_F)$, we have constructed the system of coordinates $A\mapsto(Z(A),Z^*(A))$. In the same way, for the binary equation $\widetilde{F}$ (by using a suitable dicritical foliation $\mathcal{T}_{\widetilde{F}}$ such that the separatrices $\widetilde{S}_j$ of $\mathfrak{F}_{\widetilde{F}}$ and the mirrors $l^{\pm}_{\widetilde{F}}$ of $I_{\widetilde{F}}$ are among its leaves) we construct symmetric foliations $\mathfrak{F}^{*}_{\widetilde{F}}=I_{\widetilde{F}}(\mathfrak{F}_{\widetilde{F}})$ and $\mathcal{T}_{\widetilde{F}}^*=I_{\widetilde{F}}(\mathcal{T}_F)$ and introduce coordinates $(\widetilde{Z},\widetilde{Z}^*)$ on $\mathcal{M}_{\widetilde{F}}$. Now the action of the required holomorphic diffeomorphism $\widehat{\mathcal{H}}$ can be described as follows: to a point $A\in\mathcal{M}_F$ it assigns the point $\widetilde{A}\in\mathcal{M}_{\widetilde{F}}$ with the same coordinates:
$$
\begin{equation*}
\widehat{\mathcal{H}}(A)=\widetilde{A} \quad\Longleftrightarrow\quad (Z(A),Z^{*}(A))=(\widetilde{Z}(\widetilde{A}),\widetilde{Z}^*(\widetilde{A})).
\end{equation*}
\notag
$$
Remark 18. The above construction of the map $\widehat{\mathcal{H}}$ has plenty of drawbacks. First of all, the charts $(Z,Z^*)$ and $(\widetilde{Z},\widetilde{Z}^*)$ we have defined are multivalued, so the map $\widehat{\mathcal{H}}$ can turn out to be multivalued. Second, it is unclear after all where $\widehat{\mathcal{H}}$ is defined (for example, the construction does not work at points in the separatrix sets $S_F$ and $S^*_F=I_F(S_F)$). Third, there can be problems related to the lack of transversality of $\mathfrak{F}_F$ and $\mathfrak{F}^*_F$ (which certainly occurs at points in the mirrors $l^{\pm}_F$: see Lemma 5, and at points in $\mathbb{P}$: the punctured sphere $\mathring{\mathbb{P}}$ is the common leaf of these foliations); this lack of transversality means that the systems of coordinates constructed from the pair of foliations is degenerate. However, all these drawbacks can be fixed. Moreover, we will actually construct $\widehat{\mathcal{H}}$ locally (in a neighbourhood of each point on $\mathbb{P}$) by appropriate normalization constructions; as for the above ‘definition’ of $\widehat{\mathcal{H}}$, we will use it only to glue a single diffeomorphism $\widehat{\mathcal{H}}$ defined in a full neighbourhood of $\mathbb{P}$ from these ‘pieces of $\widehat{\mathcal{H}}$’. 12.3. The consistency of the definition of $\widehat{\mathcal{H}}$ in a neighbourhood of the base point $a^+$On $\mathcal{M}_F$ we work in the chart $\tau_1=(x,p)$, $y=f_1(x,p)$. The point $a^{+}\in\mathbb{P}$ is nonsingular for the foliations $\mathfrak{F}_F$ and $\mathfrak{F}^*_F$ and lies in the domain of $\tau_1$ (see assertion 3 of Lemma 5). Consider a small bidisc $U_{+}$ with centre $a^{+}$ (which contains no singular points of $\mathfrak{F}_F$ or $\mathfrak{F}^*_F$). In $U_{+}$ we distinguish a single-valued branch of the function $Z$ (which we also denote by $Z$) by assuming that all paths $\gamma_a$ (used in its definition) are line segments. Then $Z=Z(x,p)$ is a first integral for the foliation $\mathfrak{F}_F$ (so that $(x,p)\mapsto(Z(x,p),p)$ is a straightening diffeomorphism for $\mathfrak{F}_F$). In a similar way we define a (single-valued) first integral $Z^*=Z^*(x,p)$ for $\mathfrak{F}^*_F$; then We note the main properties of $Z$ and $Z^*$. Since $Z(0,p)=Z^*(0,p)=0$, we have the following.$1^{\circ}$. $Z(x,p)=xA(x,p)$ and $Z^*=xA^*(x,p)$ for some holomorphic functions $A$ and $A^*$ in $U_{+}$; in addition, Let $A(x,p)=A_0(p)+O(x)$ and $A^*(x,p)=A_0^*(p)+O(x)$ as $x\to 0$. Then from § 8 we deduce the following property.$2^{\circ}$. The functions $A_0$ and $A_0^*=A_0\circ i$ are distinct from zero in $U_+$ and are determined by the linear part of the equation $F$. Let $p=\alpha_{+}(x)$ be the equation of the mirror $l^{+}_{F}$: Then $\alpha_{+}(0)=a^{+}$, and $A$ coincides with $A^{*}=A\circ I_F$ on $l^{+}_{F}$ (because $l^{+}_{F}$ is the mirror of $I_F$). At points in $l^{+}_F$ leaves of $\mathfrak{F}_F$ (that is, level curves of $Z$) are tangent to ‘verticals’ $\{x=\mathrm{const}\}$. However, the involution $I_F\colon (x,p)\mapsto(x,\widetilde{p}(x,p))$ preserves ‘verticals’, so the tangency between level curves of $Z$ and $Z^{*}$ at points in $l^{+}_{F}$ has order at least three. Hence the following equality holds.$3^{\circ}$. $A^{*}(x,p)-A(x,p)=c(x)\cdot(p-\alpha_{+}(x))^3+O((p-\alpha_{+}(x))^4)$ as $p\to\alpha_{+}(x)$ for some function $c(x)$ (holomorphic in $(\mathbb{C},0)$). Note that for a linear binary differential equation, $\alpha_{+}$ is a constant function: $\alpha_+(x) \equiv a^{+}$. Integrating the corresponding linear differential equations ${dx}/{dp} = xQ_1(p)$ and ${dx}/{dp}=xQ_1^*(p)$ we obtain that $c(x)$ is also a constant in the linear case: $c\equiv c_0$. It follows from (8.13) (taking the conditions of genericity into account) that this constant is distinct from zero. Repeating the same arguments for a nonlinear binary equations (so that $\alpha_+$ and $c$ are not necessarily constants, but $\alpha_+(0)=a^+$ and $c(0)=c_0$) we obtain the following property. $4^{\circ}$. $c(0)=c_0\ne0$. Lemma 16. Let the functions $Z$, $Z^{*}$, $\alpha_{+}(x)$ and $c(x)$ satisfy conditions $1^{\circ}$–$4^{\circ}$ above. Then (in a neighbourhood of the point $(x=0,p=a^{+})$) there exists a holomorphic local diffeomorphism $\Phi\colon (x,p)\mapsto(X,P)$ that is identity on the line $\{x=0\}$ and takes $(Z,Z^{*})$ to a pair of functions $(Z_0,Z^{*}_{0})$ such that $Z_0(X,P)=X$ and $Z^*_0(X,P)=Xq(P)$ , $q(P)={A_{0}^{*}(P)}/{A_0(P)}$: Proof. (1) We can assume that $\alpha_{+}(x)\equiv a^{+}$ (this can be achieved by the substitution $(x,p)\mapsto(x,p-\alpha_{+}(x)+a^{+})$).
(2) We can also achieve that $A(x,p)\equiv 1$ by making the substitution $(x,p)\mapsto(Z(x,p),p)$, which is invertible by $1^{\circ}$ and $2^{\circ}$. After it the function $A_{0}$ becomes equal to $1$, $A_{0}^{*}(p)$ to $q(p)$; the function $c(x)$ in $3^{\circ}$ is replaced by $\widetilde{c}(x)={c(x)}/{A_{0}(a^{+})}$, but the condition $\widetilde{c}_{0}:=\widetilde{c}(0)\ne0$ is preserved. (3) In the case when $\alpha_{+}\equiv a^{+}$, $A\equiv 1$ and $A^{*}(x,p)=q(p)+O(x)$ as $x\to0$ set $X(x,p)=Z(x,p)$; then (12.6) reduces to the equation However, from $3^{\circ}$ we obtain
$$
\begin{equation*}
A^{*}(x,p)=1+\widetilde{c}(x)(p-a^{+})^3+O(p-a^{+})^4, \qquad p\to a^{+},
\end{equation*}
\notag
$$
where $\widetilde{c}(x)=\widetilde{c}_0+O(x)$ and $\widetilde{c}_0\ne 0$. Therefore, for some holomorphic function $b(x,p)$, where $b(x,a^{+})\equiv0$ and $b'_p(0,a^{+})=\sqrt[3]{\widetilde{c}_0}\ne0$. Hence, in particular, for $q(p)=A^{*}(0,p)$ we obtain where $\beta(p)=b(0,a^{+})$. In particular, $\beta(a^{+})=0$ and $\beta '(a^{+})\ne0$. Then equation (12.7) assumes the following form: Therefore, it suffices to ensure that Since the function $\beta$ is invertible (in a neighbourhood of $a^{+}$), we can find $P$: It remains to observe that $P(0,p)=\beta^{-1}(b(0,p))=\beta^{-1}(\beta(p))=p$, and therefore
The proof is complete. Now it is very easy to show that $\widehat{\mathcal{H}}$ is well defined in a neighbourhood of $a^{+}$. Namely, let $\Phi$ be the holomorphic diffeomorphism from Lemma 16, which takes the pair of functions $(Z,Z^{*})$ to the normal form $(x,xq(p))$ and is constructed from the pair of foliations $(\mathfrak{F}_F,\mathfrak{F}^{*}_F)$. Note that the function $q(p)$ is determined by the linear part of $F$. Let $\widetilde{\Phi}$ be a similar holomorphic diffeomorphism (constructed from the foliations $\mathfrak{F}_{\widetilde{F}}$ and $\mathfrak{F}^{*}_{\widetilde{F}}$) that reduces the pair $(\widetilde{Z},\widetilde{Z}^{*})$ to the normal form $(x,x\widetilde{q}(p))$. However, the binary equations $F$ and $\widetilde{F}$ have the same linear part by assumption, so $q\equiv\widetilde{q}$. Hence the holomorphic diffeomorphism $\widetilde{\Phi}^{-1}\circ\Phi$ takes the pair of functions $(Z,Z^{*})$ to $(\widetilde{Z},\widetilde{Z}^{*})$. This is how we have defined the map $\widehat{\mathcal{H}}$, so $\widehat{\mathcal{H}}$ is well defined in a neighbourhood of $a^+$:
$$
\begin{equation*}
\widehat{\mathcal{H}}=\widetilde{\Phi}^{-1}\circ\Phi.
\end{equation*}
\notag
$$
Remark 19. Since the diffeomorphism in Lemma 16 is identity on the line $\{x=0\}$, the same holds for $\widehat{\mathcal{H}}$: Remark 20. Let $A$ and $A^{*}$ be points in the bidisc $U_{+}$ such that $A^{*}=I_F(A)$; then $Z^{*}(A)=Z(A^{*})$. Hence the diffeomorphism $\widehat{\mathcal{H}}$ takes $(A,A^{*})$ to a pair $(\widetilde{A},\widetilde{A}_{*})$ such that ${\widetilde{Z}^{*}(\widetilde{A})=\widetilde{Z}(\widetilde{A}_{*})}$. However, for $\widetilde{A}^{*}=I_{\widetilde{F}}(\widetilde{A})$ we also have $\widetilde{Z}^{*}(\widetilde{A})=\widetilde{Z}(\widetilde{A}^{*})$, so the points $\widetilde{A}^{*}$ and $\widetilde{A}_{*}$ have the same coordinates $(\widetilde{Z},\widetilde{Z}^{*})$. Unfortunately, this does not mean that these points coincide: leaves of the foliations $\mathfrak{F}_{\widetilde{F}}$ and $\mathfrak{F}^{*}_{\widetilde{F}}$ that pass through close points in the mirror $l^{+}_{\widetilde{F}}$ have in general three common points (in a neighbourhood of the mirror) by $3^{\circ}$. However, the points do coincide notwithstanding by (12.8): $\widetilde{A}^{*}=\widetilde{A}_{*}$, so the map constructed also conjugates the involutions $I_F$ and $I_{\widetilde{F}}$ (in a neighbourhood of $a^{+}$):
$$
\begin{equation*}
\widehat{\mathcal{H}}\circ I_F=I_{\widetilde{F}}\circ\widehat{\mathcal{H}}.
\end{equation*}
\notag
$$
Remark 21. Note also that $\widehat{\mathcal{H}}(l^{+}_F)=l^{+}_{\widetilde{F}}$ (only on these lines do we simultaneously have $Z=Z^{*}$ and $\widetilde{Z}=\widetilde{Z}^{*}$). Finally, for our choice of parameters on these transversals the map $\widehat{\mathcal{H}}$ preserves the $x$-coordinates of points on them. 12.4. The consistency of the definition of the map $\widehat{\mathcal{H}}$ in a neighbourhood of $a^{-}$Let $\gamma\subset\mathbb{P}_F$ be a path connecting the points $a^{+}$ and $a^{-}$. Let $U_{-}$ be a bidisc with centre $a^{-}$ that does not contain singular points of the foliation (as before, we are working in the chart $\tau_1$: the point $a^{-}$ lies in its domain; see Lemma 5). We select a single-valued branch $Z_{\gamma}$ of the multivalued function $Z$ by taking as $\gamma_{a}$ (in its definition) only ‘composite’ paths of the form $\gamma_a=\gamma\circ\widetilde{\gamma}_a$, where $\widetilde{\gamma}_a$ is the line segment connecting $a^{-}$with $a$. We call $Z_{\gamma}$ the ‘analytic continuation’ of the function $Z$ (or more precisely, of its element defined in § 12.3) along $\gamma$. Let $\gamma^{*}=I_F(\gamma)$; then $\gamma^{*}\subset\mathbb{P}^{*}_F$, $\gamma^{*}$ connects the point $a^{-}$ with $a^{+}$ (both are fixed points of the involution $I_F$). In a similar way we define the ‘analytic continuation’ $Z^{*}_{\gamma^{*}}$ of the function $Z^{*}$ along $\gamma^{*}$. Note that $Z_{\gamma}$ and $Z^{*}_{\gamma^{*}}$ coincide on $l^{-}_{F}$. In fact, if $\widehat{\gamma}$ is a lift of $\gamma$ to a leaf $L$ of $\mathcal{F}_F$, $A_{+}\in l^{+}_{F}$ is the initial point of $\widehat{\gamma}$ and $A_{-}\in l^{-}_{F}$ is the terminal point of $\widehat{\gamma}$, then $\widehat{\gamma}^{*}=I_F(\widehat{\gamma})$ is the lift of $\gamma^{*}$ to the leaf $L^{*}=I_F(L)$ of the symmetric foliation $\mathfrak{F}^{*}_{F}$, its initial point is $A_{+}^{*}=I_F(A_{+})=A_{+}$, and the terminal point is $A_{-}^{*}=I_F(A_{-})=A_{-}$. But this means that $Z_{\gamma}$ and $Z^{*}_{\gamma^{*}}$ coincide on $l^{-}_{F}\colon Z_{\gamma}(A_{-})=Z(A)=Z^{*}_{\gamma^{*}}(A_{-})$ for $A_{-}\in\gamma^{-}_{F}$. Thus, for the definition of $\widehat{\mathcal{H}}$ we could as well use the transversal $l^{-}_{F}$ in place of the original $l^{+}_{F}$. Now repeating all arguments from § 12.3 we can show that $\widehat{\mathcal{H}}$ is well defined in a neighbourhood of $a^{-}$ and that analogues of Remark 19 and (the first half of) Remark 21 hold here. 12.5. The consistency of the definition of $\widehat{\mathcal{H}}$ is a neighbourhood of a general nonsingular point $a$1. Let $a\in\mathbb{P}_F\cap\mathbb{P}^{*}_F$ be a nonsingular point for both $\mathfrak{F}_F$ and $\mathfrak{F}^{*}_F$ in the domain of $\tau_1$ that is distinct from $a^{\pm}$. Let $U_{a}$ be a bidisc with centre $a$ not containing singular points of $\mathfrak{F}_F$ ant $\mathfrak{F}^{*}_F$ and lying fully in the domain of $\tau_1$. We select a path $\gamma\subset\mathbb{P}_F$ connecting $a^{+}$ with $a$ and distinguish a single-valued branch $Z_a$ of the multivalued function $Z$ (the analytic continuation of $Z$ along $\gamma$) similarly to § 12.4; then $Z_a$ is holomorphic in $U_a$. Let $l\subset\mathbb{P}_F$ be a path connecting $a^{+}=a^{+}_F$ with the point $p=0$, $T_0=\{p=0\}$ be a transversal to $\mathbb{P}_F$ at $p=0$, $x$ be a parameter on $T_0$ and $\Delta_{+}\colon (l^{+}_F,a^{+}_F)\to(T_0,0)$ be the correspondence map from § 7 for $\mathfrak{F}_F$. Let $J_{+}$ be the first integral of $\mathfrak{F}_F$ from § 12.3: $J_+:=Z$; then $Z_a=\Delta^{-1}_{+}\circ J_{+}$. Taking Remark 14 into account, from this we obtain a representation for $Z_a$: where
$$
\begin{equation*}
\begin{gathered} \, \Phi_a(x,p)=\Phi^{0}_a(p)+O(x),\qquad x\to0, \\ \Phi^{0}_a(p)=\int_{a^{+}}^{p} f(t)\,dt\quad\text{and} \quad f(t)=-\frac{\mathcal{D}(t)}{P_2(t)P(t)}, \end{gathered}
\end{equation*}
\notag
$$
and the integral is taken over the path used in the definition of $Z_a$. Let $\gamma_*\subset\mathbb{P}^*_F$ be a path connecting $a^{+}$ with $a$ and $Z^*_a$ be the corresponding analytic continuation of $Z^{*}$. For $Z^*_a$ (in view of Remark 15) we obtain a similar representation:
$$
\begin{equation*}
\begin{gathered} \, Z_a^*(x,p)=x\cdot \exp(\Phi_{a*}(x,p)),\qquad \Phi_{a*}(p)=\Phi^{0}_{a*}(p)+O(x), \qquad x\to0, \\ \Phi_{a*}^0(p)=\int_{a+}^{p}f\circ i(t)i'(t)\,dt, \end{gathered}
\end{equation*}
\notag
$$
where the integral is now taken over the path in the definition of $Z_a^*$. Let $Q(x,p)=\Phi_{a*}(x,p)-\Phi_a(x,p)$; then $Q(x,p)=q(p)+O(x)$ as $x\to0$, where $q(p)=\Phi^0_{a*}(p)-\Phi^{0}_a(p)$, and we have $q'(p)=f(p)-f\circ i(p)\cdot i'(p)$, and in accordance with (8.13), $q'(a)\ne0$ because $a\ne a_F^{\pm}$. Lemma 17. There exists a local holomorphic diffeomorphism $H\colon (x,p)\mapsto(X,P)$ defined in a neighbourhood $(\mathbb{C}^2,(0,a))$, which is equal to the identity on the line $\{x=0\}$ and takes the pair of function $(Z_a,Z^*_a)$ to the standard pair $(X,X \exp q(P))$. Proof. Let $H_1\colon (x,p)\mapsto(X=Z_a(x,p),p)$; then $(Z_a,Z_a^*)=(X,Z_1)\circ H_1$, where $Z_1(X,p)=X\cdot\exp(\Phi_1(x,p))$ and $\Phi_1(x,p)=q(p)+O(x)$ as $x\to0$. For the substitution $H_2\colon (X,p)\mapsto(X,P)$, $P=\beta(X,p)$, that takes $(X,Z_1)$ to the standard pair we obtain the equation Since $q'(a)\ne0$, $q$ is invertible, so as $P$ we can take the function $P=P(x,p)=q^{-1}(Q(x,p))$. But $Q(x,p)=q(p)+O(x)$ as $x\to0$, so $P(x,p)=p+O(x)$ as $x\to0$.
The proof is complete. Now we repeat the same construction for the binary equation $\widetilde{F}$ (for the same choice of the curves $\gamma$ and $\gamma_*$). Since the linear part of $F$ and $\widetilde{F}$ coincide by assumption and $q$ in Lemma 17 is determined by the linear part of the equation $F$, Lemma 17 ensures the existence of a local diffeomorphism $\widetilde{H}$ reducing the pair $(\widetilde{Z}_a,\widetilde{Z}_a^*)$ to the same normal form. Hence the diffeomorphism $\widetilde{H}^{-1}\circ H$ takes $(Z_a,Z_a^*)$ to $(\widetilde{Z}_a,\widetilde{Z}_a^*)$, is equal to the identity on the line $\{x=0\}$ and, in addition, has the identity linear part (see Remark 22). This shows that $\widehat{\mathcal{H}}$ is well defined: in a neighborhood of a general point $a$ we take just this construction for $\widehat{\mathcal{H}}$. 2. In the above verification we produced an explicit expression for the diffeomorphism $\widehat{\mathcal{H}}:=\widetilde{H}^{-1}\circ H$; but in the construction of these $H$ and $\widetilde{H}$ we used certain auxiliary curves $\gamma$ and $\gamma_*$. Let $\gamma'\subset\mathbb{P}_F$ be another curve connecting $a^{+}$ with $a$ and $Z'_a$ be obtained by the analytic continuation of $Z$ along it; then $Z'_a=\delta\circ Z_a$, where $\delta$ is the monodromy transformation of the foliation $\mathfrak{F}_F$ corresponding to the path $\gamma\circ(\gamma')^{-1}$. In a similar way, if $\gamma'_*\subset\mathbb{P}_F^*$ is a curve connecting $a^{+}$ with $a$, and $Z_a^{*'}$ is the analytic continuation of $Z^*$ along $\gamma'_*$, then $Z^{*'}_a=\delta^*\circ Z^*_a$, where $\delta^*$ is the monodromy transformation of $\mathfrak{F}^*_F$ corresponding to $\gamma_*\circ(\gamma'_*)^{-1}$. The functions corresponding to the binary equation $\widetilde{F}$ change in just the same way (by left multiplication by the relevant monodromy transformations $\widetilde{\delta}$ and $\widetilde{\delta}^*$). However, the foliations $F$ and $\widetilde{F}$ have the same monodromy group (see (12.3)), so $\delta=\widetilde{\delta}$. Moreover, since the involution $I_F$ acts identically on $l^+_F$, it follows from the definition of $\widetilde{\mathfrak{F}}^*_F$ that the monodromy transformation of $\mathfrak{F}^*_F$ corresponding to $i(\gamma)$, $\gamma\in\pi_1(\mathbb{P}_F,a^{+})$, is the same as the monodromy transformation of $\mathfrak{F}_F$ corresponding to $\gamma$. The same holds for the pair of foliations $\mathfrak{F}_{\widetilde{F}}$ and $\mathfrak{F}^*_{\widetilde{F}}$. Then the monodromy groups of $\mathfrak{F}_{\widetilde{F}}$ and $\mathfrak{F}^*_{\widetilde{F}}$ coincide too, and therefore $\delta^*=\widetilde{\delta}^*$. Hence the corresponding diffeomorphisms $H$ and $\widetilde{H}$ in Lemma 17 change in the same way (by multiplication by the ‘diagonal’ diffeomorphism $(x,x^*)\mapsto(\delta(x),\delta^*(x^*))$ on the left). Thus the composition $\widetilde{H}^{-1}\circ H$ does not change, which means the required consistence (independence of $\mathcal{H}$ on the choice of auxiliary curves). 3. For points $a\in\mathbb{P}_F\cap \mathbb{P}^*_F$ outside the domain of $\tau_1$ we can show the consistency of the definition of $\mathbb{H}$ in a similar way. On the other hand it can be deduced simpler: there exists a natural symmetry between $x$ and $y$ (and accordingly, between $p$ and $q=p^{-1}$); furthermore, all assumptions of genericity imposed above on binary equations are also symmetric in this case. In view of this symmetry it follows from parts 1 and 2 that $\mathcal{H}$ is well defined at these points. 12.6. Consistency at singular points of $\mathfrak{F}_F$Let $(x=0,p=p_j)$ be one of the three singular points of $\mathfrak{F}_F$ ($j=1,2$ or $3$). It follows from §§ 4 and 5 that in a neighbourhood of this point $\mathfrak{F}$ is the foliation by phase curves of the vector field $\widehat{v}_F=xc\,{\partial}/{\partial x}+W(x,p)\,{\partial}/{\partial p}$, where, using the notation from these subsections, we have
$$
\begin{equation*}
\begin{gathered} \, W(x,p)=W_0(p)+O(x),\qquad x\to0, \\ W_0(p)=-\frac{P_2(p)P(p)}{\mathcal{D}(p)}. \end{gathered}
\end{equation*}
\notag
$$
In addition, $W_0(p_j)=0$, and the characteristic exponent $\lambda=\lambda_j=W'_0(p_j)$ satisfies $\operatorname{Im}\lambda\ne0$ (by Condition G4). By Poincaré’s theorem this field is linearizable: in a neighbourhood of $L:=(0,p_j)$ there exists a diffeomorphism $H_1\colon (x,p)\mapsto(u,v)$ that takes this field to the linear field $u\,{\partial}/{\partial u}+\lambda v\,{\partial}/{\partial v}$. This diffeomorphism can be selected so that $u= x$ and $v=v(x,p)=v_0(p)+o(x)$, $x\to0$, where $v_0(p)$ is a linearizing diffeomorphism for the restriction $W_0(p)\,{\partial}/{\partial p}$ of $\widehat{V}_F$ to the line $\{x=0\}\colon v'_0(p)w_0(p)=\lambda v_0(p)$ such that $v'_0(p_j)=1$. Note that these conditions define $H_1$ uniquely. By Condition G7 the point $(0,p_j)$ is nonsingular for $\mathfrak{F}^*_F\colon p_j\in\mathbb{P}^*_F$. Let $Z^*_{p_j}$ be the analytic extension of $Z^*$ along a curve $\gamma^*\subset\mathbb{P}^*_F$ connecting $a^{+}$ with $p_j$. Let $J=Z^*_{p_j}\circ H^{-1}_1$; then
$$
\begin{equation}
J(u,v)=u\cdot K(u,v),\qquad \text{where } K(u,v)=K_0(v)+O(u), \quad u\to0.
\end{equation}
\tag{12.9}
$$
Note that $K_0(v)$ is determined by the linear part of the binary equation $F$, and (as $(0,p_j)$ is a smooth point of $\mathfrak{F}^*_F$ and $p_j\ne a^{\pm}$) Lemma 18. A local diffeomorphism $H\colon (u,v)\mapsto(\widehat{u},\widehat{v})$ equal to the identity on the lines $\{u=0\}$ and $\{v=0\}$ and preserving the linear field $L$ ‘orbitally’ (that is, taking it to a proportional field) reduces $J$ to the normal form $J_0(\widehat{u},\widehat{v})=\widehat{u}\cdot K_0(\widehat{v})$: Proof. Let $\mu=\lambda^{-1}$; we seek $H\colon (u,v)\mapsto(\widehat{u},\widehat{v})$ in the form $\widehat{u}=u\cdot(g(u,v))^{\mu}$, $\widehat{v}=v\cdot g(u,v)$ for some function $g$ satisfying $g(0,0)=1$ (such a diffeomorphism certainly preserves the field $L$ ‘orbitally’ because its first integral $j=uv^{-\mu}$ does not change: $j(\widehat{u},\widehat{v})=j(u,v)$). Then (12.11) has the form $uK(u,v)=ug^{\mu}\cdot K_0(gv)$, and therefore It follows from (12.9) that (12.12) holds at the point $u=0$, $v=0$, $g=1$. However, ${\partial}/{\partial g}[g^{\mu}K_0(gv)]\big|_{u=v=0,g=1}=\mu\ne0$; hence equation (12.12) is solvable in a neighbourhood of the point $u=v=0$ (with respect to the variable $g$) by the implicit function theorem. Moreover, for $u=0$ and $g=1$ equality (12.12) holds for all $v$, so as the implicit function is unique, it follows that $g(0,v)\equiv1$. Hence $H$ acts trivially on the lines $\{u=0\}$ and $\{v=0\}$.
The proof is complete. We repeat the same construction (using, in particular, the same curve $\gamma^*\subset\mathbb{P}^*_F= \mathbb{P}^*_{\widetilde{F}}$) for the binary equation $\widetilde{F}$ and construct the corresponding diffeomorphisms $\widetilde{H}_1$ and $\widetilde{H}$. Let $H_2=H_1\circ H$, $\widetilde{H}_2=\widetilde{H}_1\circ\widetilde{H}$ and $\mathcal{H}_j=\widetilde{H}_2^{-1}\circ H_2$. By construction $\mathcal{H}_j$ is defined in a neighbourhood $U_j$ of $(0,p_j)$, takes it to another neighbourhood $\widetilde{U}_j$ of the same point, conjugates $Z^*_{p_j}$ with $\widetilde{Z}^*_{p_j}$ ($\widetilde{Z}^*_{p_j}\mathcal{H}_j=Z^*_{p_j}$) and takes the foliation $\mathfrak{F}|_{U_j}$ to $\mathfrak{F}_{\widetilde{F}}|_{\widetilde{U}_j}$. As the linear parts of $F$ and $\widetilde{F}$ coincide, it also follows that $\mathcal{H}_j$ is the identity diffeomorphism on the line $\{x=0\}$, and it has the identity linear part at points on this line. Let $D_j=U_j\cap\mathring{\mathbb{P}}$ and $D_*=D_j\setminus\{x=0,\,p=p_j\}$. In § 12.5, for each point $(0,a)\in D_*$ we constructed a diffeomorphism $\widehat{\mathcal{H}}_a$ (in a neighbourhood of this point) with just the same properties. These diffeomorphisms coincide on the intersection of their domains of definition (are analytic continuation of one another). Hence the system of these diffeomorphisms $\widehat{\mathcal{H}}_a$ forms a diffeomorphism $\mathring{\mathcal{H}}$ (with the same property) of a neighbourhood of the punctured disc $D_*$. Consider the diffeomorphism $\Phi=\widetilde{H}_2\circ\mathring{\mathcal{H}}\circ H^{-1}_2$; its domain of definition $U_*$ contains a subdomain of the form $U^{\circ}_*=\{(u,v)\colon |u|<\varepsilon_0,\,\varepsilon_1<|v|<\varepsilon_2\}$, $\varepsilon_0,\varepsilon_1,\varepsilon_2>0$. The diffeomorphism $\Phi$ maps to itself the foliation $\mathfrak{F}_L$ corresponding to the linear field $L$, preserves the function $J_0$: is equal to the identity map on the line $\{u=0\}$, and has the identity linear part at points in the annulus $D_0 = \{u = 0,\,\varepsilon_1 < |v| < \varepsilon_2\}$. Let $\Phi=(\Phi_1,\Phi_2)$, $\Phi_1(u,v)=ug_1(u,v)$ and $\Phi_2(u,v)=vg_2(u,v)$. Then $g_1=1$ and $g_2=1$ at all points in $D_0$. Since $\Phi$ takes $\mathfrak{F}_L$ to itself, it follows that $\Phi'\cdot L=L\circ\Phi\cdot K$ for some nontrivial function $K$:
$$
\begin{equation*}
\begin{gathered} \, \Phi'_{1u}u+\Phi'_{1v}\lambda v=\Phi_1 K, \\ \Phi'_{2u}u+\Phi'_{2v}\lambda v=\lambda\Phi_2 K. \end{gathered}
\end{equation*}
\notag
$$
Hence
$$
\begin{equation}
\frac{ug'_{1u}+\lambda v g'_{1v}}{g_1}=\frac{ug'_{2u}+\lambda v g'_{2v}}{\lambda g_2}.
\end{equation}
\tag{12.14}
$$
Let $G_1=\log g_1$ and $G_2=\log g_2$; since $g_1=g_2=1$ on $D_0$, both logarithms are well defined (at least in a neighbourhood of $\overline{D_0}$). Let $\psi=\lambda G_1-G_2$; then it follows from (12.14) that in a neighbourhood of $\overline{D_0}$. The function $\psi$ is holomorphic in a neighbourhood of $\overline{D_0}$; we expand it in a series: $\psi(u,v)=\sum_{k=0}^{\infty}\sum_{l\in\mathbb{Z}}\psi_{kl}u^k v^l$; then $\psi_{00}=0$. Substituting this expansion into (12.15), since the expansion is unique and $\operatorname{Im} \lambda\ne0$, we obtain $\psi_{kl}=0$ for all $(k,l)\ne(0,0)$. Hence $\psi\equiv 0$, so that $G_2=\lambda G_1$ and $g_2=g_1^{\lambda}$ in a neighbourhood of the annulus $\overline{D_0}$. Then (12.13) implies that However, then for all $(u,v)\in U_*^0$ such that $u\ne0$. By continuity this equality also holds for $u=0$. In addition,
$$
\begin{equation}
K_0(v)\ne0\quad \text{and}\quad g_1(0,v)=1 \quad\text{on } \overline{D_0}.
\end{equation}
\tag{12.17}
$$
Equation (12.16) has the trivial solution $g_1\equiv1$; at all points in the annulus $D_0$ the assumptions of the implicit function theorem hold by (12.17). By the uniqueness of the implicit function $g_1\equiv1$, so that $g_2\equiv1$; hence $\Phi=\mathrm{id}$, and therefore $\mathring{\mathcal{H}}=\mathcal{H}_j$. Thus, the diffeomorphism $\mathring{\mathcal{H}}$ extends to a full neighbourhood of the point $(x=0,p=p_j)$; this shows that the definition of $\mathring{\mathcal{H}}$ in § 12.2 is consistent. 12.7. Consistency at singular points of the foliation $\mathfrak{F}^*_F$Since singular points of $\mathfrak{F}^*_F$ have the form $(0,p^*_j)$, $p^*_j=i(p_j)$, these points are smooth for the foliation $\mathfrak{F}_F$ (as follows from Condition G7). Hence the proof of the consistency of the definition (that is, of the extension of $\mathcal{H}$ to the singular points of $\mathfrak{F}^*_F$) is quite similar to § 12.6. 12.8. The end of the proof of Theorem 2In §§ 12.2–12.7 we constructed a diffeomorphism $\widehat{\mathcal{H}}$ that (in particular) takes the foliation $\mathfrak{F}_F$ to $\mathfrak{F}_{\widetilde{F}}$. As shown in § 12.3, this diffeomorphism also conjugates the involutions $I_F$ and $I_{\widetilde{F}}$ (at least, in a neighbourhood of the point $a^{+}$). As all objects are analytic, it follows from the uniqueness theorem that $\widehat{\mathcal{H}}$ also conjugates these involutions in a full neighbourhood of the sphere $\mathbb{P}$. However, then (by Lemma 11) $\widehat{\mathcal{H}}$ descends to a local holomorphic diffeomorphism conjugating the binary equations $F$ and $\widetilde{F}$. Theorem 2 is proved. § 13. Formal equivalence of binary differential equations and their monodromy groupsThe following two definitions are formal analogues of the corresponding ‘analytic’ definitions in §§ 1 and 11. Definition 8. We say that a binary differential equation $F:=ap^2+2bp+c=0$ is formally equivalent to a binary differential equations $\widetilde{F}:=\widetilde{a}p^2+2\widetilde{b}p+\widetilde{c}=0$ if for some formal change of coordinates $\widehat{H}\colon (x,y)\mapsto(X,Y)$ and some formal series $\widehat{K}$ with nontrivial free term the following quadratic forms are equal: Then we say that the binary differential equations $\widetilde{F}$ is obtained from the binary equations $F$ by the change of variables $\widehat{H}$ and multiplication by $\widehat{K}$. Definition 9. We say that two finitely generated groups $G=\langle \Delta_1,\Delta_2\rangle $ and $\widetilde{G}=\langle \widetilde{\Delta}_1,\widetilde{\Delta}_2\rangle $, $G,\widetilde{G}\subset \operatorname{Diff}(\mathbb{C},0)$, are formally equivalent if there exists a formal change of variables $\widehat{h}(z)=\sum_{k=1}^{\infty}h_k z^k$, $h_1\ne0$, conjugating their generators: $\widehat{h}\circ\Delta_j=\widetilde{\Delta}_j\circ\widehat{h}$, $j=1,2$. Lemma 19. If two binary differential equations are formally equivalent, then their monodromy groups are too. Proof. Let the two binary differential equations $F$ and $\widetilde{F}$ be formally equivalent: $\widetilde{F}$ is obtained from $F$ by a formal change of coordinates $\widehat{H}$ and multiplication by a series $\widehat{K}$. We can assume without loss of generality that the linear parts of $F$ and $\widetilde{F}$ coincide and that $\widehat{H}'(0,0)=E$ and $K(0,0)=1$ (this can be achieved by making a linear change of coordinates and multiplying by a constant in advance). Let $H_N$ and $K_N$ be the $N$th partial sums of the formal series $\widehat{H}$ and $\widehat{K}$, respectively. Assume that $F_N$ can be obtained from the equation $F$ by an (analytic) change of variables $H_N$ and multiplication by $K_N$. Then the $N$-jet of $F_N$ coincides with the $N$-jet of $\widetilde{F}$.
Note that $\mathbb{P}_{F_{N}}=\mathbb{P}_F=\mathbb{P}_{\widetilde{F}}$ for all $N$. For some point $a\in\mathbb{P}_F$ fix generators $\gamma_1$ and $\gamma_2$ of the fundamental group $\pi_1(\mathbb{P}_F,a)$, and fix a transversal $(T,a)$ to $\mathbb{P}$ at $a$ (taking, as usual, the $x$-coordinate as a parameter on $T$). As generators of the monodromy groups $G:=G_F=\langle \Delta_1,\Delta_2\rangle $, $G_N:=G_{F_N}=\langle \Delta_1^N,\Delta_2^N\rangle$ and $\widetilde{G}:=G_{\widetilde{F}}=(\widetilde{\Delta}_1,\widetilde{\Delta}_2)$ we take the monodromy transformations of the relevant foliations that are calculated for the generators $\gamma_1$ and $\gamma_2$ and the transversal $(T,a)$. By Lemma 14 the groups $G$ and $G_N$ are analytically equivalent: there exists a local holomorphic diffeomorphism $h_N$ conjugating their generators:
$$
\begin{equation}
h_N\circ\Delta_j=\Delta_j^N\circ h_N, \qquad j=1,2.
\end{equation}
\tag{13.2}
$$
Moreover, the $N$-jet of $h_N$ is fully determined by the $N$-jet of the diffeomorphism $H_N$ and the $N$-jet of the equation $F$. However, the jets if the $H_N$ stabilize, hence the same holds for $F_N$, and also (since the $N$-jet of the monodromy transformation is determined by the $N$-jet of the binary differential equation: see Lemma 6) for the corresponding monodromy groups and their generators: for $M > N$ the $N$-jet of the germ $\Delta_j^N$ coincides with the $N$-jet of $\Delta_j^M$ (and that of $\widetilde{\Delta}_j$). Hence, as $N\to\infty$, the germs $\Delta_j^N$ converge to $\widetilde{\Delta_j}$ (coefficient-wise), and the germs $h_N$ converge to some (formal, in general) map $\widehat{h}$. Taking the limit in (13.2) we obtain the required (formal) equivalence of the groups $G$ and $\widetilde{G}$.
The proof is complete. § 14. Proof of Theorem 1Now everything is ready for the proof of Theorem 1, although we need one generic condition more. As shown in § 10, this condition is indeed satisfied for generic binary differential equations (see Lemma 9). Let $F$ and $\widetilde{F}$ be formally equivalent binary differential equations that satisfy all of Conditions G1–G8 of genericity. We can assume without loss of generality that the linear parts of $F$ and $\widetilde{F}$ coincide. Since $F$ and $\widetilde{F}$ are formally equivalent, their monodromy groups $G_F$ and $G_{\widetilde{F}}=\widetilde{G}$ are too (Lemma 19). However, these groups are unsolvable (condition G8). Hence these groups are analytically equivalent by the CMR-theorem (see [25]). Then the binary differential equations $F$ and $\widetilde{F}$ are analytically equivalent by Theorem 2. Theorem 1 is proved. 
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\Bibitem{VorChe25} |
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