Let $(M,\omega)$ be a sympletic manifold and $\{ \cdot, \cdot \}$ the corresponding Poisson-bracket. Assuming $M$ is completely integrable w.r.t $f=f_1$, so we find $n = \frac{1}{2}\dim M$ functions $f_1, \dots , f_n \colon M \to \mathbb{R}$ such that they are functionally independent and mutually Poisson-commute on an open and everywhere dense subset of $M$. We write $F := (f_1, \dots, f_n) \colon M \to \mathbb{R}^n$.
For my question I'll assume that the functions $f_1, \dots, f_n$ are functionally independent everywhere on $M$. So $F$ is a submersion and the levelsets of $F$ are lagrangian submanifolds (if I'm not mistaken). So the levelsets of $F$ define a Lagrangian Foliation of $M$.
If we now consider some additional functions $g_1, \dots, g_k$ on $M$ such that $\{f_i,g_j\}=\{g_i,g_j\}=0$, what do we know about the levelsets of $(F,g_1, \dots, g_k) \colon M \to \mathbb{R}^{n+k}$. Do we know, that the levelsets give us the same foliation as given by the levelsets of $F$?
