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 $\alpha$,s 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 orbits.

 All quadratic vector fields  with center are classified as  follows: 

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.](https://www.researchgate.net/publication/246484433_Integrability_of_plane_quadratic_vector_fields)


**Question:**

>Are there  a  family of (polynomial) Riemannian  metrics $g_{\alpha}$ on the punctured plane(after removing singularities) with  Gaussian curvature $\kappa_{\alpha}$ with the following properties:

>The  solutions of $X_{\alpha}$ are geodesics of $g_{\alpha}$.Moreover $\kappa_{\alpha}$,  is  not  zero except at a  finite  number of  algebraic curves transverse to  $X_{\alpha}$. 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,](https://en.wikipedia.org/wiki/Hilbert%27s_sixteenth_problem)  would  be  finite.


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

http://mathoverflow.net/questions/160945/limit-cycles-as-closed-geodesicsin-negatively-curved-space