According to "ON THE CONVEX PFAFF-DARBOUX THEOREM OF EKELAND AND NIRENBERG", by Bryant. See also @bryant2013exterior Theorem 3.1.
Theorem. Let $\omega$ 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 $\varphi=(w^1,\ldots,w^n)$ such that
$$ \varphi^*(\omega)=a(dw^1+w^2 dw^3+\ldots+w^{2k} dw^{2k+1}) $$with $a$ a nonvanishing function.
$\blacksquare$
This expression is called the normal form of the 1-form.
Corollary. If $\omega$ a smooth 1-form on an $n$-manifold $M$ and with constant Pfaff rank $k$ then it has integral manifolds of dimension $n-(k+1)$.
$\blacksquare$
Proof. Consider the submanifolds given by $w^1=C_1, w^3=C_2,\ldots,w^{2k+1}=C_{k+1}$$\blacksquare$ According to @olver86 page 390 it is not true in the infinite dimensional case.________________________________________
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Author of the notes: Antonio J. Pan-Collantes
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