Elliptic Schlesinger system and Painlevé VI

We consider an elliptic generalization of the Schlesinger system (ESS) with positions of marked points on an elliptic curve and its modular parameter as independent variables (the parameters in the moduli space of the complex structure). This system was originally discovered by Takasaki (hep-th/9711095) in the quasi-classical limit of the SL(N) vertex model. Our derivation is purely classical. ESS is defined as a symplectic quotient of the space of connections of bundles of degree 1 over the elliptic curves with marked points. The ESS is a non-autonomous Hamiltonian system with pairwise commuting Hamiltonians. The system is bi-Hamiltonian with respect to the linear and introduced here quadratic Poisson brackets. The latter are the multi-colour form of the Sklyanin–Feigin–Odesski classical algebras. The ESS is the monodromy independence condition on the complex structure for the linear systems related to the flat bundle. The case of four points for a special initial data is reduced to the Painleve VI equation in the form of the Zhukovsky–Volterra gyrostat, proposed in our previous paper.