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.