Let $G$ be a Lie group acting Hamiltonian on some real analytic symplectic manifold $(M, \omega)$, with an $G$-equivariant momentum map $\Phi \colon M \to \mathfrak{g}^*$.

Assuming I can find polynomials $f_1, \dots, f_n \colon \mathfrak{g}^* \to \mathbb{R}$, such that $f_1 \circ \Phi, \dots, f_n \circ \Phi \colon M \to \mathbb{R}$ are functionally independent in at least one point in $M$. Is it now possible to conclude, that $\Phi^*f_1, \dots \Phi^*f_n$ are functionally independent on some open and dense subset in $M$? Are the functions $\Phi^*f_j$ for instance real analytic?