Canonical parameterization of coadjoint orbits of $GL(N,C)$ with complicated Jordan structure and isomonodromic deformation equations

I will demonstrate that (co)adjoint orbit of $GL(N,C)$ has a structure of a symplectic fibration. It can be used for the construction of the rational Darboux coordinates on the orbit. Isomonodromic deformation equations are defined on the symplectic quotient of the product of such orbits. The iteration procedure for the solving of the momentum-level equation and the simultaneous factorization with respect to the diagonal $GL(N,C)$-action will be presented. The method works for a wide class of matrices. The isomonodromic deformations of the Fuchsian equation with the rank-one traceless matrix-residues (their Jordan forms consist of one $2\times 2$ Jordan block with zero diagonal and $(N-2)\times (N-2)$-dimension zero block) will be considered as an example.