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}(https://mathoverflow.net/q/160945/36688). 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   metric with negative curvature on the punctured plane making all  solutions of the  Van der Pol vector field into 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