Classification of the centers, their cyclicity and isochronicity for the generalized quadratic polynomial differential systems (Q1029111)

Consider the system \[ \dot z= (\lambda+ i)z+ (z\overline z)^{{d-2\over 2}}(Az^2+ Bz\overline z+ C\overline z^2),\tag{1} \] where \(z= x+ iy\), \(d\geq 2\) is an arbitrary even positive integer, \(\lambda\in\mathbb{R}\), \(A,B,C\in\mathbb{C}\). Theorem 1. For \(d\geq 2\) even, system (1) has a center at the origin if and only if one of the following conditions holds: (c.1) \(\lambda= 2A+ \overline B= 0\), (c.2) \(\lambda= \text{Im}(AB)= \text{Im}(A^3C)= \text{Im}(\overline B^3C)= 0\), (c.3) \(\lambda= B= 0\) and \(d= 2\), (c.4) \(\lambda= A- 2\overline B= |C|- |B|= 0\), \(\text{Im}(A^3C)\neq 0\) and \(d= 2\). Theorem 8. System (1) with \(d\geq 4\) even has an isochronous center at the origin if and only if one of the following conditions holds: (d.1) \(\lambda= 0\), \(B=\overline A\), \(C= 0\), (d.2) \(\lambda= 0\), \(B= (2- d)\overline A/d\), \(C= 0\), (d.3) \(\lambda= 0\), \(B= 0\), \(C= -\overline A^2/A\), \(A\neq 0\) and \(d= 4\), holds. It is also considered the cyclicity problem of the singular point \(z= 0\) of the system (1).