Perturbation theory of the quadratic Lotka-Volterra double center

We revisit the bifurcation theory of the Lotka-Volterra quadratic system \begin{eqnarray} X_0 :\left{\begin{aligned} \dot{x}=& - y -x^2+y^2 ,\ \dot{y}= &;;;;x - 2xy \end{aligned} \right. \end{eqnarray} with respect to arbitrary quadratic deformations. The system $X_0$ has a double center, which is moreover isochronous. We show that the deformed system $X_0$ can have at most two limit cycles on the finite plane, with possible distribution $(i,j)$, where $i+j\le 2$. Our approach is based on the study of pairs of bifurcation functions associated to the centers, expressed in terms of iterated path integrals of length two.