On the reductions and classical solutions of the Schlesinger equations

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×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′,m′) having n′ < n or m′ < m