Alexei Akhmetshin School of Mathematics & Statistics, University of Sydney Zero-curvature equations on algebraic curves. Field analog of the Calogero-Moser system Wednesday 31th, March 14:05-15:55pm, Carslaw Building Room 359. Following the recent work of Krichever we ouline the Hamiltonian theory of zero-curvature equations with spectral parameter on a compact Riemann surface. These equatiions can be understood as commuting flows of an infinite-dimensional field analogs of the Hitchin systems. They are explicitly parametrized with the help of so-called Tyurin parameters for stable holomorphic vector bundles. Our main example is the case of the elliptic curve with a puncture that leeds to the field elliptic Calogero-Moser (CM) system. We present the Lax pair for this system and establish its connection to the Kadomtsev-Petviashvili (KP) equation. Namely, we consider elliptic families of solutions of the KP equation, such that their poles satisfy a constraint of being balanced. We show that the dynamics of these poles is governed by a reduction of the field elliptic CM system. A wide class of solutions to the field elliptic CM system is constructed by showing that any N-fold branched cover of an elliptic curve gives rise to an elliptic family of solutions of the KP equation with balanced poles.