Hamiltonian torus actions and integrable systems

Hamiltonian torus actions as integrable systems

For a symplectic manifold $(M,\omega )$ of dimension $2n$, there is a 1-1 relationship between free, locally free, or effective Hamiltonian ${\mathbb{T}}^n$ actions and completely integrable systems.

\text{Hamiltonian torus actions} \end{Bmatrix} \leftrightarrow \{\text{Completely integrable systems}\} $$