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What is called Liouville integrability is one formalization of the notion of classical integrable system in physics.
A physical system is called Liouville integrable if it admits canonical coordinates and canonical momenta which are generated from the flow of a maximal set of commuting Hamiltonians.
A physical system given by a phase space symplectic manifold $(X, \omega)$ and equipped with a Hamiltonian $H_0 \in C^\infty(C)$ (generating time evolution) is said to be Liouville integrable or to be an integrable system in the sens of Liouville if
there are $(dim X -1)$ other Hamiltonian functions $\{H_i\}_{i = 0}^{dim X -1}$
such that
these all commute under the Poisson bracket with each other;
the flow of the corresponding Hamiltonian vector fields generates a polarization of $(X, \omega)$.