This question is inspired by this recent one and this one; I hope it's not too elementary.

Let $M$ be a (closed) smooth manifold and $X$ a vector field on $M$. Fix any Riemannian metric $g$ on $M$ and let $X^{\flat}=g(X,\cdot)$ be the metric-dual differential form of $X$.

If $dX^{\flat}=0$, what can be said about the trajectories of $X$?