For a Hamiltonian action on symplectic manifold, we have the notion of level sets defined by $\mu^{-1}(c)$ for some $c \in \mathfrak{g}^*$. Locally, a level set is a submanifold with tangent space precisely the intersection of the kernels of the 1-forms $d\mu_\xi$ for every $\xi \in \mathfrak{g}$.
I would like to know if we can generalize this idea of level sets to symplectic actions. Obviously, I wouldn't be able to give each level set a corresponding "value", since the Hamiltonian/Moment map might not be integrable (For example, the orbits of the $S^1$ action on the torus $S^1 \times S^1$ that rotates the first degree are "level sets" in the sense I'm looking for, but $\iota_X(\omega)$ is not integrable). I don't have an exact definition for what I'm looking for.
If I assume that the action is free and 1-dimensional ($\mathbb{R}$-action or $S^1$-action), the Frobenius theorem immediately gives me a foliation of the manifold to (immersed) submanifolds with tangent spaces the kernel of $dH$. That gives a nice generalization to the notion of level sets, but I run into problems when attempting to generalize this to non-free actions.
Every Hamiltonian action on a compact symplectic manifold has fixed points, so even if the action is 1-dimensional, $dH$ must be degenerate at some point. That means that we can't expect to have a foliation of the manifold to level sets in general (as can be seen for $S^2$ with the usual $H=h$). Thus, we're looking for something a bit more general.
I know that this question is not well defined, but I am open to hearing thoughts of how (or why not) to generalize level sets to symplectic actions.
For symplectic circle actions, there is the notion of a generalised moment map to $S^1$ (constructed by McDuff). In particular you have a foliation by hyperfurfaces on which the symplectic form is the zero-set of some one form (given by $dH$ by some map to $H : M \rightarrow S^1$).
The first example of a symplectic non-Hamiltonian with fixed points action was given by Mcduff, she also described the generalised Moment map $\tilde{H} :M \rightarrow S^1$ explicitly. Some of the fixed point components of the example are Riemann surfaces of high genus. I think it was also in the same paper that she proves that non-Hamiltonian, symplectic circle actions have a generalised moment map.
Later, Tolman constructed a symplectic, non-Hamiltonian circle action with non-empty and finite fixed point set.