Given a distribution $D$ on $M$, a submanifold $N\subseteq M$ such that
$$ T_q N=D_q. $$is called an integral submanifold of $D$.
An integral manifold of a distribution $D$ is a submanifold $N\subseteq M$ such that
$$ T_q N \subseteq D_q. $$We say it is locally maximal if for every $q\in N$ and a neighbourhood $U$ of $q$, $S\cap U$ is not contained in an integral manifold of bigger dimension.
The distribution $D$ always posses integral manifolds of dimension 1 (curves), but need not to posses integral manifolds of $\mbox{rank}(D)$. Even more, it can have several locally maximal manifolds through the same point, even of different dimensions. But if $D$ is involutive distribution then posses locally maximal integral submanifolds of $\mbox{rank}(D)$.
Since Pfaffian systems are a kind of dual to distributions, they have integral manifolds, also.
An integral manifold is a submanifold immersion
$$ \iota: N\to M $$such that $\iota^*(\varphi)= 0$ for all $\varphi \in \mathcal{P}$.
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Author of the notes: Antonio J. Pan-Collantes
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