Frobenius theorem (dual version)

Version 1

See @bryant2013exterior page 24.
Perhaps the simplest exterior differential systems are those whose differential ideal $I$ is generated algebraically by forms of degree one. They are called completely integrable Pfaffian systems. Let the generators be $α_{1}, \dots, α_{n - r}$ , which we suppose to be linearly independent. The condition that $I$ is closed gives:

(F) d α_{i} \equiv 0 \mod α_{1}, \dots, α_{n - r}, 1 \leq i \leq n - r .

This condition $(F)$ is called the Frobenius condition. A differential system

α_{1} = \dots = α_{n - r} = 0

satisfying $(F)$ is called completely integrable.

Geometrically, the $α$ 's span at every point $x \in M$ a subspace $W_{x}$ of dimension $n - r$ in the cotangent space $T_{x}^{*} (M)$ or, equivalently, a subspace $W_{x}^{⊥}$ of dimension $r$ in the tangent space $T_{x}$ . Such data is known as a distribution.

Theorem
Let $I$ be a differential ideal having as generators the linearly independent forms $α_{1}, \dots, α_{n - r}$ of degree one, so that the condition $(F)$ is satisfied. In a sufficiently small neighborhood, there is a coordinate system $y_{1}, \dots, y_{n}$ such that $I$ is generated by $d y_{r + 1}, \dots, d y_{n}$ .
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Version 2

@warner page 75, Theorem 2.32.
Theorem
Let $I \subset E^{*} (M)$ be a differential ideal locally generated by $d - p$ independent 1-forms. Let $m \in M$ . Then there exists a unique maximal, connected, integral manifold of $I$ through $m$ , and this integral manifold has dimension $p$ .
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