Let $(M,\omega)$ be a smooth symplectic manifold of dimension $2n$, and let $F=(f_1,\dots,f_n):M\to\mathbb R^n$ be a smooth map such that the functions $f_i$ Poisson commute and $dF_x:T_xM\to\mathbb R^n$ has rank $n$ at every point under consideration. Let $c\in\mathbb R^n$ be a regular value of $F$, and suppose that the fiber $L=F^{-1}(c)$ is compact and connected.

Then there are an open neighbourhood $W\subset\mathbb R^n$ of $c$, the [connected component](/page/Connected%20Component) $V\subset F^{-1}(W)$ containing $L$, an [open set](/page/Open%20Set) $U\subset\mathbb R^n$, and a diffeomorphism $\Phi:V\to U\times\mathbb T^n$ with coordinate functions $(I_1,\dots,I_n,\theta_1,\dots,\theta_n)$ such that $F|_V:V\to W$ is a smooth fibration by compact connected Lagrangian tori,

\begin{align*} \omega=\sum_{i=1}^n dI_i\wedge d\theta_i. \end{align*}

Moreover, for each $j\in\{1,\dots,n\}$ there exists a smooth function $h_j:U\to\mathbb R$ such that

\begin{align*} f_j=h_j(I_1,\dots,I_n) \end{align*}

on $V$.