There is a question that arises, while I'm trying to understand Guillemin & Sternbergs paper "On collective complete integrability according to the method of Thimm".
Assume $M$ is a symplectic manifold of dimension $2n$. Assume $G$ is a Liegroup, $\mathfrak{g}$ be the Liealgebra and $\mathfrak{g^*}$ the corresponding dual. Endow the $\operatorname{Ad}^*_G$ orbits of $G$ with the Kirillov-Kostant-Souriau symplectic form. By $\mathcal{O}$ we will denote a maximal dimensional $G$-orbit in $\mathfrak{g^*}$ under the $\operatorname{Ad}_G^*$ action. Let $\dim \mathcal{O} =2k$.
Assume $G$ acts on $M$ in a hamiltonian fashion, with $\operatorname{Ad}^*_G$-equivariant momentum map $\Phi \colon M \to \mathfrak{g}^*$ and such that $\Phi(M) =W$ is a submanifold.
Assume we have functions $f_1, \dots, f_k \colon W \to \mathbb{R}$ such that $\{f_i,f_j\}=0$ and such that they are functionally independent on the maximal dimensional $G$-orbits.
(So $f_1|_\mathcal{O}, \dots, f_k|_\mathcal{O} \colon \mathcal{O} \to \mathbb{R}$ are functionally independet, so $d(f_1|_\mathcal{O}) \wedge \dots \wedge d(f_k|_\mathcal{O}) \neq 0$)
Now we know, that the levelsets of $f_1|_\mathcal{O}, \dots, f_k|_\mathcal{O} \colon \mathcal{O} \to \mathbb{R}$ give us a foliation on $\mathcal{O}$, which is Lagrangian on $\mathcal{O}$ and so it is coisotropic. Since the levelsets are coisotropic submanifolds of $\mathcal{O}$ and $\Phi$ is a Poisson-map, the levelsets of $\Phi^*f_1|_\mathcal{O}, \dots, \Phi^*f_k|_\mathcal{O}$ give us a coisotropic foliation of $\Phi^{-1}(\mathcal{O})$.
If we now assume, that the $G$-orbits on $M$ are $\textbf{coisotropic}$ submanifolds, we can conclude that $\ker d_x\Phi$ is isotropic for all $x \in M$ and that we get a foliation of $M$, such that the maximal dimensional leaves are lagrangian submanifolds of $M$.
Now the authors say, that if we find $n-k$ many functions $h_1, \dots h_{n-k} \colon W \to \mathbb{R}$ whose common level surfaces are the $\operatorname{Ad}^*_G$-orbits in $W$, then the $n$ functions $\Phi^*f_1, \dots,\Phi^*f_k, \Phi^*h_1, \dots \Phi^*h_{n-k} \colon M \to \mathbb{R}$ give us a completely integrable system. That means, that the functions Poisson-commute and that they are functionally independent on $M$.
As the momentum map is a Poisson-map, it is clear that the functions Poisson-commute. But why should they be functionally independent?
Am I missing some crucial point of the momentum map for this?
Edit: Made a mistake in one assumption. We are assuming that the $G$-orbits are coisotropic and not isotropic submanifolds.
Edit2: If we write $F := (f_1, \dots, f_k)$ and assume that $\Phi$ has clean intersection with the levelsets ${F|_\mathcal{O}}^{-1}(c)$, i.e. $(d_x\Phi)^{-1}\left(T_\alpha ({F|_\mathcal{O}}^{-1}(c))\right) =T_x\left(\Phi^{-1}({F|_\mathcal{O}}^{-1}(c))\right)$, then I'm able to prove that $\Phi^*f_1, \dots,\Phi^*f_k, \Phi^*h_1, \dots \Phi^*h_{n-k} \colon M \to \mathbb{R}$ give us a completely integrable system.
But do I really have to assume that I have a clean intersection or is this somehow given by the momentum map`?