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 surrounded 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 geodesics? 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 metric on the phase space. This situation is discussed in the following MO posts and item 5 of page 3 of the this preprint.

Finding a 1-form adapted to a smooth flow

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

Limit cycles as closed geodesics(in negatively or positively curved space)

3more comments