Polynomial first integrals for weight-homogeneous planar polynomial differential systems of weight degree 3 (Q878498)

All planar polynomial systems \[ \dot{\mathbf x}={\mathbf P}({\mathbf x}), \quad {\mathbf x}\in{\mathbb C}^2 \] of weight-homogeneous degree three are classified into six types. Among them, all systems having a polynomial first integral are selected.