Bifurcation of limit cycles and integrability conditions for 6-parameter families of polynomial vector fields of arbitrary degree

We characterize the center variety and the maximum order of any focus of the 6-parameter family of real planar polynomial vectors given, in complex notation, by ż=iz+Azn1z̄j1+Bzn2z̄j2+Czn3z̄j3, where A,B,C∈C∖{0}, (n1,j1)≠(n2,j2)≠(n3,j3), nk+jk>1 for k=1,2,3n1+j1=n2+j2=n3+j3, |1−n3+j3|=|1−n2+j2|≠|1−n1+j1| and (1−n1+j1)(1−n2+j2)≠0.