Classification of Isomonodromy Problems on Elliptic Curves

We consider the isomonodromy problems for flat $G$-bundles over punctured elliptic curves $\Sigma_\tau$ with regular singularities of connections at marked points. The bundles are classified by their characteristic classes. These classes are elements of the second cohomology group $H^2(\Sigma_\tau,{\mathcal Z}(G))$, where ${\mathcal Z}(G)$ is the center of $G$. For any complex simple Lie group $G$ and arbitrary class we define the moduli space of flat bundles, and in this way construct the monodromy preserving equations in the Hamiltonian form and their Lax representations. In particular, they include the Painlev\'e VI equation, its multicomponent generalizations and elliptic Schlesinger equations. The general construction is described for punctured curves of arbitrary genus. We extend the Drinfeld-Simpson (double coset) description of the moduli space of Higgs bundles to the case of flat connections. This local description allows us to establish the Symplectic Hecke Correspondence for a wide class of the monodromy preserving equations classified by characteristic classes of underlying bundles. In particular, the Painlev\'e VI equation can be described in terms of ${\rm SL}(2, {\mathbb C})$-bundles. Since ${\mathcal Z}({\rm SL}(2, {\mathbb C}))= {\mathbb Z}_2$, the Painlev\'e VI has two representations related by the Hecke transformation: 1) as the well-known elliptic form of the Painlev\'e VI(for trivial bundles); 2) as the non-autonomous Zhukovsky-Volterra gyrostat (for non-trivial bundles).