Steklov-Lyapunov type systems

In this paper we describe integrable generalizations of the classical Steklov– Lyapunov systems, which are defined on a certain product so(m) × so(m), as well as the structure of rank r coadjoint orbits in so(m)×so(m). We show that the restriction of these systems onto some subvarieties of the orbits written in new matrix variables admits a new r × r matrix Lax representation in a generalized Gaudin form with a rational spectral parameter. In the case of rank 2 orbits a corresponding 2×2 La x pair for the reduced systems enables us to perform a separation of variables.