Bäcklund Transformations on Coadjoint Orbits of the Loop Algebra g̃l(r)

There is a wide class of integrable Hamiltonian systems on finite-dimensional coadjoint orbits of the loop algebra g̃l(r) which are represented by r × r Lax equations with a rational spectral parameter. A reduced complex phase space is foliated with open subsets of Jacobians of regularized spectral curves. Under some generic restrictions on the structure of the Lax matrix, we propose an algebraic geometrical scheme of a discretization of such systems that preserve their first integrals and is represented as translations on the Jacobians. A generic discretizing map is given implicitly in the form of an intertwining relation (a discrete Lax pair) with an extra parameter governing the translation. Some special cases when the map is explicit are also considered. As an example, we consider a modified discrete version of the classical Neumann system described by a 2× 2 discrete Lax pair and provide its theta-functional solution.