Let $(X, \omega)$ be a symplectic manifold of dimension $2n$ and $\omega$ is an exact symplectic form i.e. $\omega = -d\alpha$. Let furthermore $M \subset X$ be a compact Lagrangian submanifold such that $\alpha = 0$ of $TX|\_{M}$. I am interested in the following questions:


1. Is there a unique polarisation defined on $X$ near $M$ which is transversal to $M$ and whose one form is $\alpha$ ? By polarisation I mean the following: A polarisation of a symplectic manifold $X$, with symplectic form $\omega$, is a smooth assignment of a Lagrangian subspace of $T\_{x}X$ to each $x \in X$ in such a way that this assignment is integrable.
2. If 1. is true, is there a symplectic diffeomorpism $\Phi$ of a neigbourhood of $M$ is $X$ with a neigbourhood of $M$ in its cotangent bundle which carries the leaves of the polarisation into the standard cotangent fibration of $T^{\*}M$ ?


Actually I know that these results are true. I would like to see the proof of them. Are there any references where I can look them up?



Polarisation in a neighbourhood of a Lagrangian submanifold
[Arnol'd](http://www.ams.org/mathscinet-getitem?mr=1768639) (p. 3314) puts it that way: 



> 
> **Weinstein's Theorem**. Some neighborhood of any Lagrangian submanifold in
>  any symplectic manifold is symplectomorphic to some neighborhood of this Lagrangian submanifold in any other symplectic manifold, for instance in its own cotangent bundle space.
> 
> 
> 


(The resulting neighborhood then has the obvious transverse polarization by fibers of the cotangent bundle.) Unless I am mistaken, Weinstein proves this in [*Symplectic manifolds and their lagrangian submanifolds*](http://www.ams.org/mathscinet-getitem?mr=286137), Theorem 6.1 and Corollary 6.2 (which he points out goes back to [Souriau](http://www.ams.org/mathscinet-getitem?mr=60290)).


