Let $M^n$ be a smooth manifold. Let $\mathcal{D}$ be a $k$-dimensional integrable distribution on $M$. Denote by $\mathcal{D}^\bot\to M$ the vector bundle which is spanned by the differentials of the set of first integrals of $\mathcal{D}$. That is, at a point $p$ we have $\mathcal{D}^\bot|_p = \left<\mathrm{d}f\ \Big|\ \forall\xi\in\mathcal{D}|_p\colon\xi(f) = 0\right>$.
Then for any point $p\in M$ there are functions $f^1_p, \ldots, f^{n - k}_p$ such that in a neighbourhood of $p$ the differential $(n - k)$-form $\omega_{(p)} := \mathrm{d}f^1_p\wedge\ldots\wedge\mathrm{d}f^{n - k}_p$ is nowhere vanishing and the vector bundle $\mathcal{D}^\bot$ equals $\left<\mathrm{d}f^1_p, \ldots, \mathrm{d}f^{n - k}_p\right>$, so $\omega_{(p)}$ is "top-dimensional".
Clearly, we have $\mathrm{d}\omega_{(p)} = 0$. Also, each differential $(n - k)$-form $\tau\in\bigwedge^{n - k}\mathcal{D}^\bot|_p$ is proportional to $\omega_{(p)}$.
As one can see, the existence of such $\omega_{(p)}$ for each $p\in M$ is a local condition. What I wonder is when a similar condition is met globally.
To be precise, the question is the following. When does there exist a nowhere vanishing $(n - k)$-dimensional differential form $\omega$ such that $\mathrm{d}\omega = 0$ and at each point $p\in M$ the form $\omega$ is proportional to $\omega_{(p)}$ (or, equivalently, $\mathrm{d}\omega = 0$ and $\forall p\in M\colon\omega|_p\in\bigwedge^{n - k}\mathcal{D}^\bot|_p$)?
This obviously always holds for $k = 0, n$ (a constant function and a volume form will suffice). However, I have not been able to think of (and prove) a criterion even for $k = 1$. In particular, I have tried using a partition of unity but have not succeeded.