On the cohomological equation of magnetic flows

We consider a magnetic flow without conjugate points on a closed manifold $M$ with generating vector field $\text{G}$. Let $h\in C^{\mathrm{\infty }}(M)$ and let $\theta$ be a smooth 1-form on $M$. We show that the cohomological equation $$\text{G}(u)=h\circ \pi +\theta$$ has a solution $u\in C^{\mathrm{\infty }}(SM)$ only if $h=0$ and $\theta$ is closed. This result was proved in \cite{DP2} under the assumption that the flow of $\text{G}$ is Anosov.