Let $X$ be a nonvanishing vector field on a Riemannian surface $(M,g)$. For $q\in M$, the geodesic curvature of the orbit of$ X$ passiing $q$ is denoted by $\kappa_g(q)$.

Definition: A non vanishing vector field $X$ on a $2$-manifold $M$ satisfies WG property (weak geodesible) if there is a Riemannian metric $g$ on $M$ such that the smooth function $\kappa_g(q).|X(q)|$ belongs to the range of the derivation operator $X(u)=X.u=du(X)$.

Remark 1: Obviousely every geodesible flow is a WG flow.

Question 1: Is the Vander Pole vector field $(V)$ bellow a WG vector field on $\mathbb{R}^2\setminus \{0\}$?$$(V)\;\;\;\begin{cases} x'=y-(x^3-x)\\y'=-x\end{cases}$$

Question 2: Is there a negative curvature Riemannian metric $g$ on the punctured plane for which $\kappa_g |V|$ lies in the range of derivation associated to $(V)$.

Proposition: If the answer to the second question is affirmative, then we obviousely have an alternative proof for the fact that $(V)$ has at most one limit cycle.

Proof: If $\gamma_1, \gamma_2$ are two limit cycles then $\int_{\gamma_i} \kappa_gds=\int_{\gamma_i} \kappa_g|V|dt=0$. Applying the Gauss Bonnete theorem to the annular Region bounded by $\gamma_1, \gamma_2$ gives us a contradiction.

Remark 2: The current post is considered as a possible resolution to difficulties appeared in the following efforts for consideration of Limit cycles as closed geodesics