Centers for generalized quintic polynomial differential systems (Q2409590)

The authors consider real planar polynomial differential systems \[ \dot z= iz+(z\overline z)^{(d-5)/2}(Az^5+ Bz^4\overline z+ Cz^3\overline z^2+ Dz^2\overline z^3+ Ez\overline z^4+ F\overline z^5),\tag{1} \] where \(z= x+iy\), \(d\geq 5\) is an arbitrary odd integer and \(A,B,C,D,E,F\in \mathbb{C}\) satisfy one of the four conditions: {\parindent=0.8cm \begin{itemize}\item[(c.1)] \(A= \text{Re}(D)= 0\), \item[(c.2)] \(A=\text{Im}(D)= 0\), \item[(c.3)] \(\text{Re}(A)= D= 0\), \item[(c.4)] \(\text{Im}(A)= D= 0\). \end{itemize}} Sufficient center conditions of (1) are obtained.