8. What’s Integrable System
In mathematics and physics, there are various distinct notions
that are referred to under the name of integrable systems.
In the general theory of differential systems, there is
Frobenius integrability, which refers to overdetermined systems.
In the classical theory of Hamiltonian dynamical systems, there is
the notion of Liouville integrability. More generally, in
differentiable dynamical systems integrability relates to the
existence of foliations by invariant submanifolds within the
phase space. Each of these notions involves an application of the
idea of foliations, but they do not coincide. There are also
notions of complete integrability, or exact solvability in the
setting of quantum systems and statistical mechanical models.