Bifurcation of limit cycles from a quadratic reversible center with the unbounded elliptic separatrix

The paper is concerned with the bifurcation of limit cycles in general quadratic perturbations of a quadratic reversible and non-Hamiltonian system, whose period annulus is bounded by an elliptic separatrix related to a singularity at infinity in the poincar'{e} disk. Attention goes to the number of limit cycles produced by the period annulus under perturbations. By using the appropriate Picard-Fuchs equations and studying the geometric properties of two planar curves, we prove that the maximal number of limit cycles bifurcating from the period annulus under small quadratic perturbations is two.