A geometric property of polynomial differential systems of degree 2\(k\) (Q1206835)

Consider the polynomial differential system in the plane \(dx/dt=P_{2k}(x,y)\), \(dy/dt=Q_{2k}(x,y)\), where \(P_{2k}(x,y)\) and \(Q_{2k}(x,y)\) are polynomials of degree \(2k\), \(k\geq 1\), in two variables \(x\), \(y\) with real coefficients, and no common factors. The author proves that the phase portrait of this differential system, mapped onto a hemisphere, can never be a center and its family of closed orbits.