On a class of cubic systems with two centers (Q1118081)

Consider the system \(\dot x=P(x,y)\), \(\dot y=Q(x,y)\) where P, Q are two real polynomials of \((x,y)\in {\mathbb{R}}^ 2\) of relatively prime degrees \(\leq k\) with one at least of degree k. It is shown that certain cubic systems with two centers and no other singular points can have a rectilinear trajectory or none, but cannot have two such trajectories. Three examples are discussed.