  We discuss geometrical aspects of different dualities in the integrable
systems of the Hitchin type and its various generalizations. It is shown that T
duality known in the string theory context is related to the separation of
variables procedure in dynamical system. We argue that there are analogues of S
duality as well as mirror symmetry in the many-body systems of Hitchin type.
The different approaches to the double elliptic systems are unified using the
geometry behind the Mukai-Odesskii algebra.
