  An old conjecture claims that commuting Hamiltonians of the double-elliptic
integrable system are constructed from the theta-functions associated with
Riemann surfaces from the Seiberg-Witten family, with moduli treated as
dynamical variables and the Seiberg-Witten differential providing the
pre-symplectic structure. We describe a number of theta-constant equations
needed to prove this conjecture for the $N$-particle system. These equations
provide an alternative method to derive the Seiberg-Witten prepotential and we
illustrate this by calculating the perturbative contribution. We provide
evidence that the solutions to the commutativity equations are exhausted by the
double-elliptic system and its degenerations (Calogero and Ruijsenaars
systems). Further, the theta-function identities that lie behind the Poisson
commutativity of the three-particle Hamiltonians are proven.
