Thanks a lot for the comments. I am interested in the case of a compact, connected Hamiltonian $T-$manifold $M$ with moment map $\mu: M \rightarrow \mathfrak{t}^*$. A theorem of Atiyah states that any non-empty preimage of $\mu$ is connected. This is closely related to the famous convexity result that $\mu(M)$ is convex in $\mathfrak{t}^*$.