Nilpotent global centers of generalized polynomial Kukles system with degree three (Q6583753)

Consider the planar polynomial system\N\[\N\begin{array}{l} \frac{{dx}}{{dt}}= y(1+a_1x+a_2x^2), \\\N\frac{{dy}}{{dt}} = -2(1+\lambda^2)x^3+(4\lambda+b_1x)xy+(b_2+b_3x)y^2+b_4y^3, \end{array}\tag{1}\N\]\Nwhere \((a_1,a_2,b_2,b_3,b_4,\lambda)\in \mathbb{R}^6\) and \(b_1>0\). The authors derive necessary and sufficient conditions for the origin of system (1) to be a global center. The behavior of system (1) at infinity is part of these conditions.