  In these lectures I consider the Hitchin integrable systems and their
relations with the self-duality equations and the twisted super-symmetric
Yang-Mills theory in four dimension follow Hitchin and Kapustin-Witten. I
define the Symplectic Hecke correspondence between different integrable
systems. As an example I 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.
