On the normal form of multimode squeezings

We describe the solution of algebraic equations for the coefficients of the normal factorization
$$ U_t=e^{i\widehat H t}=e^{s_t}e^{-\frac{1}{2}(a^\dagger,R_ta^\dagger)-(g_t,a^\dagger)}\,e^{(a^\dagger,C_t a)}\,e^{\frac{1}{2} (a,\overline\rho_t a)+(\overline f_t,a)} $$
of the unitary group $U_t$ with Hamiltonian
$$ \widehat H= \frac{i}{2}((a^\dagger,Aa^\dagger)-(a,\overline A a))+(a^\dagger,B a)+i(a^\dagger,h)-i(a,\overline h) $$
in terms of the matrices $\Phi_t$, $\Psi_t$ which define the canonical transformation of the creation-annihilation operators. Such a decomposition defines the normal symbol of squeezing and the inner products of squeezed states which are necessary for constructing a basis in a linear hull generated by a finite set of squeezed states. A new class of solvable quantum problems is related to Hamiltonians with $A$ and $B$ such that $[A\overline A,B]=0$ and $\operatorname{rank}A\overline A\ge \operatorname{rank}B$. In this case, the solution is expressed in terms of the eigenvalues of the Hermitian matrix $A\overline A-B^2$.