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.