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 $P,Q$ are polynomials of degree $2$ with $P(0,0)=Q(0,0)=0$. We define the planar algebraic curve $$C=\{(x,y) \mid yP-xQ=0\}$$

The above linked answer shows that:

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.

2) 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.


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