2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/50564We 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.Latex, 29 pages, Contribution to the Proceedings of NATO Advanced Research Workshop, Kiev, September 2000High Energy Physics - Theorytext