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. Integrability can often be traced back to the algebraic geometry of differential operators.