# Obstructions for finding a volume form for a given flow

Let $$X$$ be a non-singular $$C^\infty$$ vector field on a three manifold $$M$$. There are some obvious obstructions for finding a volume form that is preserved under the flow given by $$X$$: If $$X$$ is singular, or If there's a sphere to which $$X$$ is transverse (which implies a singularity), or if $$X$$ is transverse to a torus. Could it be that these are all the obstructions? Or is there a theorem characterizing all topological obstructions?

I have a Riemannian metric on the manifold to begin with, and I want to change the metric so that the vector field (that avoids the above obstructions) becomes divergence free, but is still smooth or at least $$C^1$$.

Thanks for any hints/answers/references!