  We calculate, in a relatively explicit way, the Hamiltonian systems which
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.
