Quadratic centers defining elliptic surfaces
The purpose of the paper is to classify all the first integrals of planar quadratic differential systems (not necessarily having a center) which define elliptic curves. In the first part, a full classification, up to an affine equivalence, is given of all elliptic foliations, that is those having a center. There are 18 reversible cases, 5 reversible Lotka-Volterra cases and 6 generic Lotka-Volterra cases in the classification. In the second part, all degenerate cases (i.e. without a center) are determined and the topology of their singular fibers is investigated. The results from the first part would be useful to add several new classes of quadratic centers whose limit cycles bifurcating under small perturbations are tractable by the technique (also known as the Abelian integrals method) previously used for the elliptic Hamiltonian cases.
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