Linear Pfaffian Systems and Classical Solutions of Triangular Schlesinger Equations

In this paper, by classical solutions we mean solutions to Fuchsian type meromorphic linear integrable Pfaffian systems $\mathrm d y=\Omega y$ on the complex linear spaces $\mathbb C^n$, $n\geq 1$, where $y(z) = (y_1(z),\dots ,y_n(z))^\top \in \mathbb C^n$ is a column vector and $\Omega $ is a meromorphic matrix differential $1$-form such that $\Omega =\sum _{1\leq i<j\leq n}J_{ij}(\beta )(z_i-z_j)^{-1}\,\mathrm d(z_i-z_j)$, with constant matrix coefficients $J_{ij}(\beta )$ depending on complex parameters $\beta =(\beta _1,\dots ,\beta _n)$. Under some constraints on the constant matrix coefficients $J_{ij}(\beta )$, the solution components $y_i(z)$, $1\leq i\leq n$, can be expressed as integrals of products of powers of linear functions; i.e., they are generalizations of the integral representation of the classical hypergeometric function $F(z,a,b,c)$. Moreover, under some additional constraints on the parameters $\beta $, the components of the solutions are hyperelliptic, superelliptic, or polynomial functions. We describe such constraints on the coefficients $J_{ij}(\beta )$ of Fuchsian type systems, as well as describe constraints on the sets of matrices $(B_1(z),\dots ,B_n(z))$ for which the nonlinear Schlesinger equations $\mathrm dB_i(z)=-\sum _{j=1,\,j\neq i}^n[B_i(z),B_j(z)](z_i-z_j)^{-1}\,\mathrm d(z_i-z_j)$ reduce to linear integrable Pfaffian systems of the type described above and have solutions of the indicated type.