A quadratic system is a polynomial vector field on the plane in the form $$X_{\alpha}\;\;\;\;\;\;\begin{cases}\dot x=P(x,y)\\ \dot y=Q(x,y) \end{cases}$$

where $P,Q$ are polynomials of degree $2$. The coefficients of $P,Q$ are denoted by $\alpha$. So the $\alpha$ in $X_{\alpha}$ is indicating to coefficients of the vector field.

A center is a singularity of this vector field which is surrounded by a band of closed orbit. All quadratic vector fields with center are classified: they correspond to a finite number of algebraic conditions in $\alpha$, These algebraic conditions are denoted by $Cent(\alpha)=0$. see "Integrability of plane quadratic vector fields" Expos. Math(1990)3-25.

Question:

Are there a family of (polynomial) Riemannian metrics $g_{\alpha}$ with the following properties:

The solutions of $X_{\alpha}$ are geodesics of $g_{\alpha}$.Moreover the gaussian curvature of $g_{\alpha}$, denoted by $\kappa_{\alpha}$, is not zero except at a finite number of algebraic curves which consist either a singular points or points at which $X_{\alpha}$ is transverse to the equation $\kappa_{\alpha}=0$. Moreover $\kappa_{\alpha}$ is identically zero if $Cent(\alpha)=0$.

If the answer would be yes, then $H(2)$, the maximum number of limit cycles of a quadratic system, would be finite.

This question is already discussed at the comment-conversation of the following post:

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