On the moment map on symplectic manifolds

We consider a connected symplectic manifold $M$ acted on by a connected Lie group $G$ in a Hamiltonian fashion. If $G$ is compact, we prove give an Equivalence Theorem for the symplectic manifolds whose squared moment map $\parallel \mu \parallel^2$ is constant. This result works also in the almost-K\"ahler setting. Then we study the case when $G$ is a non compact Lie group acting properly on $M$ and we prove a splitting results for symplectic manifolds.