Integrable system

Finite degree of freedom

It is common knowledge that every Hamiltonian system is locally integrable (away from singular points of the Hamiltonian), meaning that, in a neighborhood of each point of the $2 n$ -dimensional symplectic manifold on which the Hamiltonian vector field is defined, it is possible to find $n$ integrals of motion in involution. Moreover, they are, locally, superintegrable systems, in the sense that they have $(2 n - 1)$ first integral of motions.

Then, the question of integrability and superintegrability it is interesting only if we:

make it a global question (Liouville integrability), or
still in a local set up, restrict the class of "allowed functions", for example, to polynomial functions.

An special case consists of that systems which can be expressed with a Lax pair.

Liouville integrability

In the special setting of Hamiltonian systems, we have the notion of integrability in the Liouville sense or completely integrable system. Liouville integrability means that there exists a regular foliation (Liouville foliation) of the phase space by invariant manifolds such that the Hamiltonian vector fields associated with the invariants of the foliation span the tangent distribution.
Another way to state this is that there exists a maximal set of functionally independent Poisson-commuting invariants ${f_{i}}$ (i.e., independent functions $f_{i}$ on the phase space whose Poisson brackets with the Hamiltonian of the system, and with each other, vanish).

Infinite degree of freedom (fields)

(According to this)
Many nonlinear PDEs possess families of traveling wave, periodic and solitary wave solutions. However, the addition of integrability endows an equation with a much deeper mathematical structure. In turn, integrability provides one with corresponding mathematical methods to probe more deeply the structure of the solution space, which allows one to do something quite rare in the study of PDEs: explicitly write down a large family of physically meaningful solutions that can be observed in nature and study in detail their dynamical behavior.

In the context of PDEs, integrable systems could generally be viewed as those equations that can be obtained from an overdetermined system of linear differential equations, with the original PDE playing the role of the compatibility condition. The overdetermined linear system is the Lax pair associated to the equation. If such a Lax pair representation is known for a PDE, then many methods of analysis can be applied to study the PDE. In particular, methods such as the Inverse Scattering Method (IST), which is a generalization of the Fourier transform to non-linear PDE and is an effective solution method for initial value problems.

Related: evolution equation.