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Given a Poisson manifold $(P,\{\cdot,\cdot\})$, its characteristic distribution $\mathcal C$ is the singular tangent distribution on $M$ generated by the Hamiltonian vector fields,i.e. $$\mathcal C=\operatorname{span} \{ X_f:= \{ f,\cdot \} \mid f\in C^\infty (M) \}.$$

The Symplectic Foliation Theorem states that:

  • $\mathcal C$ is completely integrable à la Sussmann-Stefan
  • each integral manifold of $\mathcal C$ brings a unique symplectic Poisson structure such that the immersion map is a Poisson morphism.

In "The Local Structure of Poisson Manifolds" A. Weinstein attributes the Symplectic Foliation Theorem to Sophus Lie for Poisson manifolds having constant rank (citing precisely the Second Section of "Theorie der Transformationsgruppen") and to A.A.Kirillov and R.Hermann in the general case.

I would like to learn more how this problem is tackled by Lie, and if there are some modern expositions (preferibly in English) of Lie's approach to this theorem?

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