Let $S,T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ be defined through $S(x,y)=(x+1,-y)$ and $T(x,y)=(x,y+1)$. Show that if $\omega$ is a continuous and invariant ($S^*(\omega)=T^*(\omega)=\omega$) 2-form in $\mathbb{R}^2$, then there exista an invariant $C^1$ 1-form $\alpha$ such that $d\alpha=\omega$.
I managed to solve the smooth case by sending $\omega$ to the Klein Bottle and using its cohomology to create a $\beta$ that can be pulled back to the plane.
On the continuous case, I managed sufficient and necessary conditions on $\omega$ to get an $\alpha$ of the form $g \ dx$ or $g \ dy$, but these conditions don't seem to be valid for any $\omega$.
I know little of the approximations theorems so I might be digging an unecessary rabbit hole.
After (many) calculations on a PDE the conditions are:
$\omega = f \ dx \wedge dy$
$$\int_0^1 f(x,s)ds =0 \ \forall \ x$$ or $$\int_0^1 f(s,y) - f(s,-y)ds = 0 \ \forall \ y$$