Given a symplectic manifold $(M, \omega)$, a smooth function $H:M \to \mathbb{R}$, we can show that for a regular level set $H^{-1}(\lambda)$, if there is a free action $\mathbb{S}^1 \times H^{-1}(\lambda) \to H^{-1}(\lambda)$, then the quotient $H^{-1}(\lambda)/\mathbb{S}^1$ will be a symplectic submanifold of $M$.
The main problem I have though is that the proof relies on the fact that $H^{-1}(\lambda)$ is a coisotropic submanifold (since it is codimension 1), and that the orbits of $\mathbb{S}^1$ are along the null directions $T_pH^{-1}(\lambda)^\omega$. I don't understand this last claim. I understand that a coisotropic subspace contains its annihilating set, however it's not at all obvious to me why the orbits of the action must be constrained to these directions. Why is this the case? Does this draw on the theory of foliations? What is the explanation?
Ok, I was a little near-sighted in posing this question. The action of $\mathbb{S}^1$ is meant to be a Hamiltonian action meaning that $\mathbb{S}^1$ acts by symplectomorphisms, so that $(t,p) \to \psi_t(p)$ where the $\psi_t$ are $1$-periodic with $\psi_0 = Id$. This action is Hamiltonian in the sense that
$$ \frac{d}{dt} \psi_t(p) \;\; =\;\; X_H|_{\psi_t(p)} $$
for a Hamiltonian vector field $X_H$ corresponding to $H \in C^\infty(M)$. The vector field $X_H$ is in fact the $\omega$-null direction in $T_pH^{-1}(\lambda)$ since $H$ is constant along a level set and $i_{X_H}\omega |_{H^{-1}(\lambda)} = dH |_{H^{-1}(\lambda)} = 0$.