Abstract: We revisit the bifurcation theory of the Lotka-Volterra quadratic system $x' = − y − x^2 + y^2, y' = x − 2xy$, for arbitrary quadratic deformations. The system has a double center, and we first compute an associated pair of Bautin ideals. We show that the deformed system can have at most two limit cycles on the finite plane, with possible distribution $(i, j)$, where  $i + j ≤ 2$. Our approach is based on the study of pairs of bifurcation functions associated with the centers, expressed in terms of iterated path integrals of length two. This is a joint work with Lubomir Gavrilov.