Nonautonomous Hamiltonian systems related to higher Hitchin integrals

We describe nonautonomous Hamiltonian systems derived from the Hitchin integrable systems. The Hitchin integrals of motion depend on $\mathcal W$-structures of the basic curve. The parameters of the $\mathcal W$-structures play the role of times. In particular, the quadratic integrals depend on the complex structure (the $\mathcal W_2$-structure) of the basic curve, and the times are coordinates in the Teichmьller space. The corresponding flows are the monodromy-preserving equations such as the Schlesinger equations, the Painlevé VI equation, and their generalizations. The equations corresponding to the higher integrals are the monodromy-preserving conditions with respect to changing the $\mathcal W_k$-structures $(k>2)$. They are derived by the symplectic reduction of a gauge field theory on the basic curve interacting with the $\mathcal W_k$-gravity. As a by-product, we obtain the classical Ward identities in this theory.