Homogeneous Poisson structures on symmetric spaces

We calculate, in a relatively explicit way, the Hamiltonian systems that arise from the Evens-Lu construction of homogeneous Poisson structures on both compact- and noncompact-type symmetric spaces. A corollary is that the Hamiltonian system arising in the noncompact case is isomorphic to the generic Hamiltonian system arising in the compact case. In the group case, these systems are also isomorphic to those arising from the Bruhat-Poisson structure on the flag space, and hence, by results of Lu, can be completely factored.