Let $M$ be a smooth connected $d$-dimensional manifold, $E$ a vector bundle over $M$ of rank $d$. Let $\sigma \in \Omega^{d-1}(M,E)$ be a non-zero $E$-valued form on $M$ (of degree $d-1$).
I am interested in the subspace $A_{\sigma}=\{ h \in C^{\infty}(M) \, | \, \sigma \wedge dh =0\}$.
($dh$ is a real valued one form, so $\sigma \wedge dh \in \Omega^{d}(M,E)$).
$A_{\sigma}$ always contain the constant functions, so $\dim A_{\sigma} \ge 1$. Is it true that $\dim A_{\sigma} > 1$? Can $\dim A_{\sigma}$ be finite? infinite?
Can we construct an "explicit" basis for $A_{\sigma}$ (or even one non-constant function in $A_{\sigma}$) in terms of $\sigma$?
One possible strategy is to consider the relaxed equation $$\sigma \wedge \omega=0, \omega \in \Omega^1(M), \tag{1}$$
look for closed solutions, and hope they might be exact (The topology of $M$ might enter the game, of course).
Actually, we should start by making sure there is no "pointwise" obstruction to equation $(1)$:
Let $p \in M$. Does there exist $\omega_p \in T_p^*M$ such that $\sigma_p \wedge \omega_p=0$?
(This is a fiberwise question, on the level of linear algebra).
Locally $\sigma$ is given by
$$ \sum_{i,j=1}^d (-1)^{i-1}\sigma_i^j dx^1 \wedge \cdots \wedge \widehat{dx^i} \wedge \cdots \wedge dx^d \otimes e_j,$$
where $\{e_1, \cdots, e_d\}$ is a local basis for $E$. So
$$dh \wedge \sigma = \sigma_i^j h_i dx^1 \cdots dx^d \otimes e_j $$
So $\sigma \wedge dh = 0$ would mean $ \sigma \cdot h = 0$ (think as matrix multiplications). So for a generic $\sigma$, $\sigma_i^j$ is invertible on a dense subset of $M$, so $dh=0$ on a dense subset and so $dh=0$ on $M$.
Thus for a generic $\sigma$, $A_\sigma$ is one dimensional.