Given an exterior differential system defined on a manifold "M", the Cartan–Kuranishi prolongation theorem says that after a finite number of "prolongations" the system is either "in involution" (admits at least one 'large' integral manifold), or is impossible.
History.
The theorem is named after Élie Cartan and Masatake Kuranishi. Cartan made several attempts in 1946 to prove the result, but it was in 1957 that Kuranishi provided a proof of Cartan's conjecture.
Applications.
This theorem is used in infinite-dimensional Lie theory.
References.
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