Let $M$ be a differential manifold and $\Omega^p(M)$ the vector bundle of $p$-forms. My question is:
Given a differential $p$-form $\omega$, is there a differential $q$-form $\phi$ such that $d(\omega\wedge\phi)=0$?
I am excluding the trivial cases when $\omega$ is already closed or when $(q+p)$ is larger or equal to the dimension of the cotangent space at a point.
My question is a generalization of the integrating factor problem, where $\omega$ is a 1-form, $\phi$ is a function and $d(f\omega)$ should be exact and not only closed as I am requiring. In this question about the existence of integrating factor for 1-forms in two variables the answers say that the problem is difficult and still open, even in this simpler case.
I was unable to find any reference who could be of some help in answer my question, then I would really appreciate if someone can give me some directions in the literature and, if possible, discuss some special cases where a solution is or is not possible.
a) If $\omega\in\Omega^{p}(M)$ for $p$ odd, then $\omega\wedge\omega=0$, so certainly $d(\omega\wedge\omega)=0$.
b) If $\omega\in\Omega^{p}(M)$ for $p$ even, then $d(\omega\wedge d\omega)=d\omega\wedge d\omega=0$.