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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. Integrability can often be traced back to the algebraic geometry of differential operators.
==Frobenius integrability (overdetermined differential systems)==
A differential system is said to be ''completely integrable'' in the Frobenius sense if the space on which it is defined has a foliation by maximal integral manifolds. The Frobenius theorem states that a system is completely integrable if and only if it generates an ideal that is closed under exterior differentiation. (See the article on integrability conditions for differential systems for a detailed discussion of foliations by maximal integral manifolds.)
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