On the isoclines of polynomial vector fields (Q1920853)

The author considers autonomous systems (1) \(\dot x = P(x,y)\), \(\dot y = Q(x,y)\) with \(\{P(x,y), Q(x,y)\}\) a polynomial vector field. He proves the following results. Theorem 1. Assume \(P(x,y) = P_m (x,y) + P_n (x,y)\), \(Q(x,y) = Q_m (x,y) + Q_n (x,y)\) with \(m,n > 0\) and \(P_m\), \(Q_m\) and \(P_n\), \(Q_n\) homogeneous polynomials of degree \(m\) and degree \(n\) respectively. If \(m\) and \(n\) are of different parity, then at least one isocline passes through the singular point \((0,0)\). Moreover, either each straight line passing through \((0,0)\) is an isocline or there are at most \(m + n\) isoclines. Theorem 2. Let the vector field in (1) be polynomial with \((0,0)\) a singular point. Every straight line passing through this point is an isocline if and only if the field is collinear with a homogeneous vector field, i.e., there exist homogeneous polynomials \(P_k (x,y)\) and \(Q_k (x,y)\) \((k \geq 0)\) and a polynomial \(R(x,y)\) such that the representation \(P(x,y) \equiv P_k (x,y) R(x,y)\), \(Q(x,y) \equiv Q_k (x,y) R(x,y)\) holds.