TLDR: Symplectic Quotient of the whole space; not a levelset of a moment map. Using a symplectic slice. What goes wrong?
Setup
Let $G$ be a Lie Group acting smoothly, freely and properly on a symplectic Manifod $(M,\omega)$. Then the quotient $M/G$ is a smooth manifold such that $\pi: M\to G/M$ is a submersion. We want to define a sympelctic struture on $M/G$ in the case of a symplectic action. (I know about Marsden-Weinstein reduction but bear with me here)
We use a slice of the action to define a two form $\omega^\text{red}$ on $G/M$. I wonder when this two form will be symplectic.
Construction
Assume that $H\subset TM$ is a slice of the $G$-action: $H$ is an equivariant subbundle of $TM$ that is complemented to the tangentspaces of orbits. More precisely $$ H_{g.p}={\lambda_g}_\ast H_p\quad \text{and}\quad T_pM=H_p\oplus T_p(G.p) \quad\text{for }g\in G,p\in M. $$ Then one can identify tangentspaces of the quotient $M/G$ with this slice: The derivative $\pi_\ast$ restricts to an isomorphism $H_p\simeq T_{\pi(p)}(M/G)$. And any two identifications of $T_{\pi(p)}(M/G)$ only differ by left translation.
Thus if we assume that the action is symplectic we can define a two form $\omega^\text{red}$ on $G/M$ via $$ \omega_{\pi(p)}^\text{red}(\pi_\ast X,\pi_\ast Y) =\omega (X,Y) \quad\text{for every }X,Y\in H_p. $$
Question
Now $\omega^\text{red}$ is non-degenerate if the slice is symplectic, ie $H_p^{\perp_\omega}\cap H_p = 0$ for every $p\in M$. Notice that existence of a symplectic slice excludes the usual reason for the need of a moment map: $\dim M/G$ is even.
But when is $\omega^\text{red}$ closed? Could we use something along the lines $\pi^\ast \omega^\text{red}=\omega$ and $d \circ \pi^\ast=\pi^\ast \circ d$? Or do we need extra conditions on the slice? Maybe something to do with integrability
In general, it is not closed. For instance if $G=\S^1$ is a Circle acts on $C^2-{0}$, the quotient is $S^2 \times ]0, +\infty[$, and the form is $ t\omega_0$, where $\omega _0$ is the area for on $S^2$, and $t$ the second variable (the moment map). The point is that the volume of the spheres $J^{-1}(t)$ varies.(Duistermaat Eckmann). To insure that $\omega _{red}$ is closed,, you must ask the bundle to be trivial...