Limit cycles of quadratic systems and closed geodesics(Finitness of $H(2)$)

This question is inspired by this answer to the question Finding a 1-form adapted to a smooth flow.

Assume that $$V$$ is a polynomial vector field of degree $$2$$ as follows:$$\begin{cases} x'=P(x,y)\\y'=Q(x,y) \end{cases}\;\;\;\;(V)$$ where $$P(x,y),Q(x.y)$$ are polynomials of degree $$2$$ in $$x,y$$ with $$P(0,0)=Q(0,0)=0$$.

We define the planar algebraic curve $$C=\{(x,y) \mid yP-xQ=0\}$$

There is a Riemannian metric $$g$$ on $$\mathbb{R}^2 \setminus C$$ such that the trajectories of $$V$$ are geodesics with respect to the metric $$g$$. That is $$V$$ is a geodesible vector field on $$\mathbb{R}^2 \setminus C$$.

Now my question is that:

Is there a Riemannian metric $$g$$ on $$\mathbb{R}^2 \setminus C$$ with curvature $$\kappa$$ such that the following two properties hold?

1. Trajectories of $$V$$ are (unparametrized) geodesic.
1. Either the curvature $$\kappa$$ is identically zero, correspending to "center singularity", or the set $$\{(x,y)\mid \kappa(x,y)=0\}$$, is transverse to the vector field $$V$$? If the answer is yes, what is a precise formulation for this metric? (Precise formulation of the metric in terms of $$P,Q$$).

According to the method of Proof of Proposition 6.7 and 6.8 in Page 71 of the book "Geometry of Foliation" By Phillip Tondeur, it seems that a relevant(but not necessarily unique) metric we are searching for, should has the orthonormal frame $$\{\frac{1}{\chi(V)}V,E\}$$ where $$V$$ is the above quadratic vector field $$V$$, $$E$$ is an arbitrary direction parallel to the radial vector field $$x\partial_x + y\partial_y$$ and $$\chi$$ is $$\chi=\frac{1}{x^2+y^2}(ydx-xdy)$$ Notice that $$E$$ lies in the kernel of the $$1-$$ form $$\chi$$.

So it seems that one can control the number of limit cycles of a typical quadratic vector field by the number of connected components which are determined by the complement of curves $$yP-xQ=0$$ and $$\kappa=0$$. Please see the above linked answer.

Are there some other metric, not necessarily associated with the above frame $$\{\frac{1}{\chi(V)}V,E\}$$, for which the vector field $$V$$ is a geodesible vector field?Is there a conformal one?

In fact the motivation is that, we would like to find a procedure to count the number of limit cycles of a (typical) quadratic vector field $$(V)$$. A procedure based on the curvature sign and Gauss Bonnet theorem.

Please see this question for further explanation.

Remark: The initial motivation is mentioned in page 3, item 5 of this arxiv note.