  Integrable N-particle systems have an important property that the associated
Seiberg-Witten prepotentials satisfy the WDVV equations. However, this does not
apply to the most interesting class of elliptic and double-elliptic systems.
Studying the commutativity conjecture for theta-functions on the families of
associated spectral curves, we derive some other non-linear equations for the
perturbative Seiberg-Witten prepotential, which turn out to have exactly the
double-elliptic system as their generic solution. In contrast with the WDVV
equations, the new equations acquire non-perturbative corrections which are
straightforwardly deducible from the commutativity conditions. We obtain such
corrections in the first non-trivial case of N=3 and describe the structure of
non-perturbative solutions as expansions in powers of the flat moduli with
coefficients that are (quasi)modular forms of the elliptic parameter.
