Planar polynomial vector fields having first integrals and algebraic limit cycles (Q1036169)

The authors study differential polynomial systems of the form \[ \begin{cases} \dot x =P_m(x,y)+P_{m+n}(x,y) +P_{m+2n}(x,y),\\ \dot y =Q_m(x,y)+Q_{m+n}(x,y) +Q_{m+2n}(x,y), \end{cases} \] where \(P_s\) and \(Q_s\) are homogeneous polynomials of degree \(s\) in \(x\) and \(y\). They obtain a new class of Darboux integrable systems inside the family which have degenerate infinity. Some limit cycles of the systems are detected. The method is based on transformation of the system to the Abel differential equation using polar coordinates.