The centers and their cyclicity for a class of polynomial differential systems of degree 7

We classify the global phase portraits in the Poincaré disc of the generalized Kukles systems ẋ=−y,ẏ=x+axy⁶+bx³y⁴+cx⁵y²+dx⁷,which are symmetric with respect to both axes of coordinates. Moreover using the averaging theory up to sixth order, we study the cyclicity of the center located at the origin of coordinates, i.e. how many limit cycles can bifurcate from the origin of coordinates of the previous differential system when we perturb it inside the class of all polynomial differential systems of degree 7.