Isn't it a consequence of the existence of action-angle semi-global coordinates? Namely, one has that
If $F=(f_1,\dots,f_n)$ is proper, then for any lagrangian invariant torus $\Lambda$ there exists a neighborhood $U$ of $\Lambda$ that is symplectomorphic to
a neighborhood $V$ of a fiber of the projection in the cotangent bundle $T^*\mathbb{T}^n$ of the $n$-dimensional torus $\mathbb{T}^n$ (namely, $V=\{(\theta,I), |I-I_0|\leq r\}$).
It seems to me that the zero section $\{\theta=0\}$ satisfies your requirement.
EDIT: if you read french then you can find a proof of this statement in Lecture Notes by Colin de Verdière (proof of Theorem 20, pages 50-51).