Let $X$ be a geodesible non vanishing vector field on a manifold $M$. Namely there is a Riemannian structure $(M,g)$ such that all integral curves of $M$ are unparametrized geodesics of the metric $g$. {This is a beautiful non example}(https://mathoverflow.net/a/274981/36688)

Is the index of an orbit or a closed orbit (i.e the index of a geodesic or a closed geodesics ) encoded in the vector field $X$? Namely can we compute the number of conjugate points on a (closed) orbits of $X$ with information just from the vector field and nothing else?

This question could play a crucial role in investigation of the following post about a {negatively curved structure on the punctured plane for which the solution curves of the Van der Pol equation would be geodesics}(Limit cycles as closed geodesics (in negatively or positively curved space)). Because if the answer to this post is affirmative(or there are some modified way to compute the index of the closed geodesics of this metric) and we get a non zero index this would implies that there is no a negative curvature metric on the punctured plane making all solutions of the Van der Pol geodesics.

The next question: Regardless of the sign of the curvature, is there a Riemannian metric on the Punctured plane such that solution of the Vander pol equation are geodesics and there is no conjugate point at all