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Liouville integrability means having $n$ almost-everywhere independent first integrals in mutual Poisson involution.

For a Hamiltonian phase space of dimension $2n$, Liouville integrability means $n$ functionally independent conserved quantities with pairwise vanishing Poisson brackets. This is the Hamiltonian specialization of integrable systems.

Integrable systems admit unusually rich exact structure such as Lax pairs, conserved quantities, and symmetry reductions.