 Infinite-dimensional Hamiltonian integrability

For a field evolving by a Hamiltonian partial differential equation, infinite-dimensional integrability combines sufficiently rich commuting conservation laws with a mechanism such as the inverse scattering transform or a spectral coordinate system that linearizes evolution. Lax equations supply spectral invariants; Lenard-Magri recursions supply commuting Hamiltonians when their solvability conditions hold. Infinite many conserved quantities alone do not prove completeness or an infinite-dimensional action-angle theorem: the function space, boundary conditions and analytic independence all matter.