I am interested in finding conditions to impose over a one form $\alpha$ defined over a torus $\mathbb{T}^2$ to be sure that there is no function $g:\mathbb{T}^2\rightarrow\mathbb{R}$ solving the equation $$ dg\wedge \alpha + gd\alpha = 0. $$
I have seen that if $\alpha$ is closed, then it is enough to take $g\equiv c\in\mathbb{R}$ to get the equation satisfied. So necessarily $d\alpha\neq 0$.
Is there some name for this kind of equations? How can I find some condition on $\alpha$?
Remark: I have that $g\not\equiv 0$ since I actually have $g=1/f$ for some function $f:\mathbb{T}^2\rightarrow\mathbb{R}$.
This can't be done. You can always choose $g$ to be the function $g(x)=0$ for all $x \in \mathbb{T}^2$.