Reading about the geometrical theory of systems of first order pdes, I have met a result from symplectic geometry, that is easy to prove, but I am unable to give a reference for it. So my question is:



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Let $W$ be a coisotropic embedded submanifold of a symplectic manifold $(M,\omega)$.
Let $T\_W M$ be the restriction to $W$ of the tangent bundle of $M$.
In the symplectic vector bundle $(T\_W M,\omega|\_{T\_W M})$, let the symplectic complement of $TW$ be denoted by $(TW)^\perp$. 


Then $(TW)^\perp$, being a vector subbundle of $TW$ because of the coisotropy of $W$, is completely integrable.



On the Complete integrability of a tangent distribution
This is an exercise in McDuff and Salamon's *Introduction to Symplectic Topology*. In the second edition, it's exercise 3.29. I don't know if an exercise is a great reference, but I've seen them used before.


