Centers for a class of generalized quintic polynomial differential systems

We characterize the centers of the planar polynomial differential systems of degree d⩾5d⩾5 odd that in complex notation writes asż=iz+(zz¯)d-52(Az5+Bz4z¯+Cz3z¯2+Dz2z¯3+Ezz¯4+Fz¯5),where A,B,C,D,E,F∈C and either A=Re(E)=0A=Re(E)=0, or A=Im(E)=0A=Im(E)=0, or Re(A)=E=0Re(A)=E=0, or Im(A)=E=0Im(A)=E=0.