How to Characterize Gradient Vector Fields?

Let $V$ be a vector field on a smooth manifold $M$.

Are there nice conditions under which there exists a (Riemannian) metric on $M$ such that $V$ is the gradient of some smooth function on $M$?

One obstruction is that gradient vector fields have no closed integral curves (since a function is increasing on integral curves of its gradient).

An obstruction for a vector field to be a gradient vector field is it violate the Riemannian version of Thom gradient conjecture.

The famous Thom gradient conjecture (which is true in the analytic Riemannian contex) says that: Let $X$ be an analytic gradient vector field with a singular point at a given point say $0$, WLOG. Assume that a solution curve $\gamma(t)$ tends to $0$ as $t$ goes to $\infty$. Then $\lim \frac{\gamma(t)}{|\gamma(t)|}$ exist. This conjecture is proved(around 2000, in Annals of Math), furthermore its Riemannian version is true(Please google search for "Thom gradient conjecture" and "Riemannian version of Thom gradient conjecture.

So if an anlytic vector field $X$ posses an oscilatory solution tending to a singularity, then there is no any Riemannian metric which make $X$ as a gradient vector field

Please see this MO post too:

https://mathoverflow.net/a/301874/36688