On the Reductions and Classical Solutions of the Schlesinger equations

Abstract

The Schlesinger equations $S_{(n,m)}$ describe monodromy preserving deformations of order $m$ Fuchsian systems with $n+1$ poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of $n$ copies of $m\times m$ matrix algebras equipped with the standard linear Poisson bracket. In this paper we address the problem of reduction of particular solutions of ``more complicated'' Schlesinger equations $S_{(n,m)}$ to ``simpler'' $S_{(n^{\prime },m^{\prime })}$ having $n^{\prime }<n$ or $m^{\prime }<m$.

32 pages. To the memory of our friend Andrei Bolibruch