Integral manifolds: simple approach

Given a distribution $D$ on $M$, a submanifold $N\subseteq M$ such that

is called an integral submanifold of $D$.

Integral manifolds: finer approach

An integral manifold of a distribution $D$ is a submanifold $N\subseteq M$ such that

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 $\text{rank}(D)$. 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 $\text{rank}(D)$.

In the case of Pfaffian systems

Since Pfaffian systems are a kind of dual to distributions, they have integral manifolds, also.
An integral manifold is a submanifold immersion

such that ${\iota }^{\ast }(\phi )=0$ for all $\phi \in \mathcal{P}$.