The separability and dynamical $r$-matrix for the constrained flows of Jaulent-Miodek hierarchy

Abstract

We show here the separability of Hamilton-Jacobi equation for a hierarchy of integrable Hamiltonian systems obtained from the constrained flows of the Jaulent-Miodek hierarchy. The classical Poisson structure for these Hamiltonian systems is constructed. The associated $r$-matrices depend not only on the spectral parameters, but also on the dynamical variables and, for consistency, have to obey the classical Yang-Baxter equations of dynamical type. Some new solutions of classical dynamical Yang-Baxter equations are presented. Thus these integrable systems provide examples both for the dynamical $r$-matrix and for the separable Hamiltonian system not having a natural Hamiltonian form.