CLASSICAL INTEGRABLE SYSTEMS AND GAUGE FIELD THEORIES
In this review we consider the Hitchin integrable systems and their relations with the self-duality equations and twisted supersymmetric Yang-Mills theory in four dimensions. We define the symplectic Hecke correspondence between different integrable systems. As an example, we consider elliptic Calogero-Moser system and integrable Euler-Arnold top on coadjoint orbits of the group GL(N, C) and explain the symplectic Hecke correspondence for these systems.