The paper studies the cyclicity of a class of quadratic reversible systems with hyperelliptic level curves and two period annuli and proves that for the concrete studied class, at most two limit cycles may appear from each annulus, and simultaneously from both at the same time, i.e., they may only appear in configurations (2, 0), (0, 2), (1, 1), (1, 0), (0, 1) and (0, 0).
The authors study a very particular class of quadratic systems having two centers with only one parameter. The phase portraits are of the class known as Vulpe 3, with two centers, two saddles and an invariant ellipse. It is not yet proved that this phase portrait has always the described cyclicity since the initial family they propose with two parameters also may produce Vulpe 3 (and other phase portraits with centers as Vulpe 20, Vulpe 5, Vulpe23, ...). The authors say that it is a difficult task to study the isolated zeros of the Abelian integral, and thus need to reduce the study to a single parameter and fix \(a=4\). Moreover, the other parameter is restricted to \(0<b<2\) in order to obtain Vulpe 3. Curiously, if \(a<0\), we may obtain Vulpe 20 and this phase portrait is known to be able to produce the configuration of limit cycles (3,1), so it is clear that the parameter \(a\) greatly affects the problem.