In this paper we classify the phase portraits in the Poincaré disc of the centers of the generalized class of Kukles systems
symmetric with respect to the
16e50005b2b62" title="Click to view the MathML source">y-axis, and we study, using the averaging theory up to sixth order, the limit cycles which bifurcate from the periodic solutions of these centers when we perturb them inside the class of all polynomial differential systems of degree
4.