Elliptic solutions to difference non-linear equations and nested Bethe ansatz equations

We outline an approach to a theory of various generalizations of the elliptic Calogero-Moser (CM) and Ruijsenaars-Shneider (RS) systems based on a special inverse problem for linear operators with elliptic coefficients. Hamiltonian theory of such systems is developed with the help of the universal symplectic structure proposed by D.H. Phong and the author. Canonically conjugated action-angle variables for spin generalizations of the elliptic CM and RS systems are found.