Pfaff-Darboux theorem

According to "ON THE CONVEX PFAFF-DARBOUX THEOREM OF EKELAND AND NIRENBERG", by Bryant. See also @bryant2013exterior Theorem 3.1.

Theorem. Let $ω$ be a smooth 1-form on an $n$ -manifold $M$ and with constant Pfaff rank $k$ . For every $x \in M$ there exists an open neighbourhood $U$ with a coordinate system $φ = (w^{1}, \dots, w^{n})$ such that

φ^{*} (ω) = a (d w^{1} + w^{2} d w^{3} + \dots + w^{2 k} d w^{2 k + 1})

with $a$ a nonvanishing function.
$◼$

This expression is called the normal form of the 1-form.

Corollary. If $ω$ a smooth 1-form on an $n$ -manifold $M$ and with constant Pfaff rank $k$ then it has integral manifolds of dimension $n - (k + 1)$ .
$◼$
Proof. Consider the submanifolds given by $w^{1} = C_{1}, w^{3} = C_{2}, \dots, w^{2 k + 1} = C_{k + 1}$ $◼$

According to @olver86 page 390 it is not true in the infinite dimensional case.