Three-particle integrable systems with elliptic dependence on momenta and theta function identities

We claim that some non-trivial theta-function identities at higher genus can stand behind the Poisson commutativity of the Hamiltonians of elliptic integrable systems, which were introduced in [1], [2] and are made from the theta-functions on Jacobians of the Seiberg-Witten curves. For the case of three-particle systems the genus-2 identities are found and presented in the Letter. The connection with the Macdonald identities is established. The genus-2 theta-function identities provide the direct way to construct the Poisson structure in terms of the coordinates on the Jacobian of the spectral curve and the elements of its period matrix. The Lax representations for the two-particle systems are also obtained.