THE CENTERS IN THE REDUCED KUKLES SYSTEM

In this paper we consider the family of the cubic systems of Kukles with the condition that one of the parameters alpha(7) is zero. Under this restriction the centre conditions were given by Kukles. The study of this family exhibits properties and issues which are important in the problem of the full classification of cubic systems with a centre. The family is formed of four strata, one of which is made up of quadratic systems and was studied before by Schlomiuk. If we consider the three strata formed by truly cubic systems, we have a first (second) stratum consisting of systems symmetric with respect to the x-axis (y-axis) and a third stratum consisting of systems with two invariant straight lines and having an elementary first integral obtained by the Darboux method. Systems in either one of the symmetric strata do not possess elementary first integrals generically. The first stratum is formed by integrable systems having a Liouvillian first integral. We show that systems in the second stratum have no Liouvillian first integral. We give the full bifurcation diagram of each stratum of truly cubic systems.