Cartan–Kähler theorem

In mathematics, the Cartan–Kähler theorem is a major result on the integrability conditions for differential systems, in the case of analytic functions, for differential ideals $$I$$. It is named for Élie Cartan and Erich Kähler.

Meaning
It is not true that merely having $$dI$$ contained in $$I$$ is sufficient for integrability. There is a problem caused by singular solutions. The theorem computes certain constants that must satisfy an inequality in order that there be a solution.

Statement
Let $$(M,I)$$ be a real analytic EDS. Assume that $$P \subseteq M$$ is a connected, $$k$$-dimensional, real analytic, regular integral manifold of $$I$$ with $$r(P) \geq 0$$ (i.e., the tangent spaces $$T_p P$$ are "extendable" to higher dimensional integral elements).

Moreover, assume there is a real analytic submanifold $$R \subseteq M$$ of codimension $$r(P)$$ containing $$P$$ and such that $$T_pR \cap H(T_pP)$$ has dimension $$k+1$$ for all $$p \in P$$.

Then there exists a (locally) unique connected, $$(k+1)$$-dimensional, real analytic integral manifold $$X \subseteq M$$ of $$I$$ that satisfies $$P \subseteq X \subseteq R$$.

Proof and assumptions
The Cauchy-Kovalevskaya theorem is used in the proof, so the analyticity is necessary.