Is there a distribution not expressible as a kernel of a list of 1-forms?

Question

Given a smooth manifold $X$ and a smooth distribution/vector-subbundle $E\subset TX$ on $X$ of codimension $k$, is it possible that there do not exist $\omega_1,\ldots,\omega_k\in \Omega^1(X)$ such that $E = \mathrm{ker}\omega_1 \cap \cdots \cap \mathrm{ker}\omega_k$?

Answer 1

The Möbius strip $$ X=[0,2\pi]\times[-1,1]/\sim, $$ with $E=\text{Span}(\partial_{x})$ works.

If there would exist $\omega\in\Omega^{1}(X)$ such that $E=\ker\omega$, then $\omega$ is of the form $$ \omega=f(x,y)dy. $$ Moreover $f(0,y)=-f(2\pi,-y)$, because $\omega$ is invariant under $(0,y)\sim(2\pi,-y)$. Restricting to the segment $[0,2\pi]\times\{0\}$, we see that $$ f(0,0)=-f(2\pi,0). $$ By the intermediate value theorem, f has a zero on $[0,2\pi]\times\{0\}$. At such a point, $\ker\omega$ has dimension 2, which is a contradiction.