Homogeneous Poisson Structures on Symmetric Spaces

We calculate, in a relatively explicit way, the Hamiltoniansystems that arise from the Evens–Lu construction of homogeneousPoisson structures on both compact- and noncompact-type symmetricspaces. A corollary is that the Hamiltonian system arising inthe noncompact case is isomorphic to the generic Hamiltoniansystem arising in the compact case. In the group case, thesesystems are also isomorphic to those arising from the Bruhat–Poissonstructure on the flag space, and hence, by results of Lu, canbe completely factored.