Quadratic perturbations of a quadratic reversible center of genus one (Q644532)

The bifurcation of limit cycles for small quadratic perturbations of the planar system with a center \[ x'=y-28x^2+32y^2,\quad y'=-x(1+32y) \] is studied. This is case (r15) in the classification of all quadratic systems whose generic complexified periodic orbits define elliptic curves [\textit{S. Gautier, L. Gavrilov} and \textit{I. D. Iliev}, Discrete Contin. Dyn. Syst. 25, No. 2, 511--535 (2009; Zbl 1178.34037)]. By using the method of Abelian integrals, it is proved that the period annulus around the center at the origin has cyclicity two. The same result was obtained earlier (by another method) in the paper of \textit{M. Grau, F. Mañosas} and \textit{J. Villadelprat} [``A Chebyshev criterion for Abelian integrals'', Trans. Am. Math. Soc. 363, No. 1, 109--129 (2011; Zbl 1217.34052)].