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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.

The nonlinear Schrödinger equation adds an amplitude-dependent interaction to Schrödinger-type wave evolution. Its Laplacian term disperses the complex wave amplitude, while the sign and form of the nonlinear term determine focusing or defocusing behavior. A one-dimensional cubic specialization is the focusing nonlinear Schrodinger equation; members of this broader family need not be integrable systems.

The second Painlevé equation, in its standard normalization. Similarity reductions of some integrable systems give equations of this form, after rescaling variables.

Hamilton's equations givesoUseThenand $P_2=p_2-(n_2/n_1)p_1$ is the conserved combination. The transformed Hamiltonian depends on $Q_1$ but not $Q_2$. Thus $H$ and $P_2$ are two independent first integrals in involution for two degrees of freedom, proving Liouville integrability.

In light-cone coordinates, the sine-Gordon equation can be written $u_{XT}=\sin u$. Its inverse scattering transform, solitons, breathers, and scaling symmetries make it an integrable system.