  We use the geometry of the moment map to investigate properties of pure
entangled states of composite quantum systems. The orbits of equally entangled
states are mapped by the moment map on coadjoint orbits of local
transformations (unitary transformations which do not change entanglement),
thus the geometry of coadjoint orbits provides a partial classification of
different entanglement classes. To achieve the full classification a further
study of fibers of the moment map is needed. We show how this can be done
effectively in the case of the bipartite entanglement by employing Brion's
theorem. In particular, we presented the exact description of the partial
symplectic structure of all local orbits for two bosons, fermions and
distinguishable particles.
