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 $(U, φ = (x_{1}, \dots, x_{n}))$ such that

D = S ({φ_{*}^{- 1} (\partial x_{1}), \dots, φ_{*}^{- 1} (\partial x_{r})})

(that is, it is completely integrable distribution).
Equivalently, the dual description of the distribution would be

D^{*} = S ({d x_{r + 1}, \dots, d x_{n}})

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

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}, \dots, c_{n})} := {q \in U : x_{i} (q) = c_{i}, for i = r + 1, \dots, 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.