  We present a description of locally equivalent states in terms of symplectic
geometry. Using the moment map between local orbits in the space of states and
coadjoint orbits of the local unitary group we reduce the problem of local
unitary equivalence to an easy part consisting of identifying the proper
coadjoint orbit and a harder problem of the geometry of fibers of the moment
map. We give a detailed analysis of the properties of orbits of "equally
entangled states". In particular we show connections between certain symplectic
properties of orbits such as their isotropy and coisotropy with effective
criteria of local unitary equivalence.
