Integrable systems from the classical reflection equation

We construct integrable Hamiltonian systems on G/K, where G is a coboundary Poisson-Lie group and K is a Lie subgroup arising as the fixed point set of a group automorphism σ of G satisfying the classical reflection equation. We show that the time evolution of these systems is described by a Lax equation, and under a factorizability assumption, present its solution in terms of a factorization problem in G. Our construction is closely related to the semiclassical limit of Sklyanin's integrable quantum spin chains with reflecting boundaries.