Assume that $P(x,y),Q(x,y)\in \mathbb{R}[x,y]$ are two polynomials of degree $2$ with $P(0,0)=Q(0,0)=0.$
Suppose  that the vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}$$ has a center singularity at the origin. This means that the origin is surounded by a band of closed orbits.

>Is there a Riemannian metric on $\mathbb{R}^2 \setminus  C$  with zero curvature such that all closed orbits of the vector field are closed geodesic?
Here $C$ is the algebraic curve $$C=\{(x,y)\mid yP(x,y)-xQ(x,y)=0\}$$


The motivation comes from the idea of consideration of "Limit cycles" of polynomial vector fields as $Closed Geodesics"  of a Riemannian meteic on the phase space.
This situation is discussed in the following MO posts and point 5 of page 3 of the following  preprint.

https://mathoverflow.net/questions/273635/finding-a-1-form-adapted-to-a-smooth-flow/273648#273648


https://mathoverflow.net/questions/275823/limit-cycles-of-quadratic-systems-and-closed-geodesicsfinitness-of-h2


https://mathoverflow.net/questions/160945/limit-cycles-as-closed-geodesicsin-negatively-or-positively-curved-space?noredirect=1&lq=1


https://arxiv.org/abs/math/0507516