Let $M$ be an orientable 3-manifold. On $M$, fix a vector field $X$. The curl of $X$ relative to the Riemannien metric $g$ and the volume form $\mu$, $\nabla_{g,\mu}\times X$, is defined by the formula $$ di_X g=i_{\nabla_{g,\mu}\times X}\mu.$$

When is it possible to choose a metric and volume form such that $$ \nabla_{g,\mu}\times X=\lambda X,$$ where $\lambda$ is a nowhere vanishing function?

There are many $X$ for which such a metric and volume form can be found. In particular, $X$ that arise as Reeb vector fields relative to some contact 1-form on $M$ are all examples (http://www.math.upenn.edu/~ghrist/preprints/beltrami.pdf).