  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 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 paper. 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.
