Frobenius theorem (local)

A rank $r$ distribution $D$ on an $n$-dimensional manifold $M$ is involutive if and only if for every $p\in M$ there exists a coordinate chart $\left(U, \varphi=(x_1,\ldots,x_n)\right)$ such that

$$ D=\mathcal{S}(\{\varphi_*^{-1}(\partial x_1),\ldots,\varphi_*^{-1}(\partial x_{r})\}) $$

(that is, it is completely integrable distribution).

Equivalently, the dual description of the distribution would be

$$ D^*=\mathcal{S}(\{dx_{r+1},\ldots,dx_n\}) $$

Poof. It uses the canonical form of commuting vector fields. $\blacksquare$

Therefore, Frobenius theorem tells us that involutive distributions have neighbourhoods in which there exist integral submanifolds crossing any given point, with the biggest dimension possible. They are described by:

$$ N_{(c_{r+1},\ldots,c_n)}:=\{q\in U: x_i(q)=c_i \text{, for } i=r+1,\ldots,n\} $$

In infinite dimension, i.e., a Banach manifold, Frobenius theorem states that a subundle of the tangent bundle is integrable iff it is involutive. See the section Banach manifolds in this link.

As a particular case we have the rank 1 distributions, that are always involutive and, therefore, always have integral manifolds called, in this case, integral lines. See canonical form of a regular vector field.