Suppose that $(M,\omega)$ is a (connected compact) symplectic manifold with a Hamiltonian $S^1$-action given by Hamiltonian $H$. I would like to find a reference for the fact that every level set of $H$ is connected. I tried to find this statement in McDuff Salamon, but could not.
Michael F. Atiyah, Convexity and commuting Hamiltonians (1982), Lemma 2.3.
Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology (2nd ed., 1998), Lemmas 5.51 and 5.54.
Michèle Audin, Torus actions on symplectic manifolds (2nd ed., 2004), Corollary IV.3.2.