On the number of invariant conics for the polynomial vector fields defined on quadrics

The quadrics here considered are the nine real quadrics: parabolic cylinder, elliptic cylinder, hyperbolic cylinder, cone, hyperboloid of one sheet, hyperbolic paraboloid, elliptic paraboloid, ellipsoid and hyperboloid of two sheets. Let Q be one of these quadrics. We consider a polynomial vector field X=(P,Q,R) in R3 whose flow leaves Q invariant. If m1 = degree P, m2 = degree Q and m3 = degree R, we say that m=(m1,m2,m3) is the degree of X. In function of these degrees we find a bound for the maximum number of invariant conics of X that results from the intersection of invariant planes of X with Q. The conics obtained can be degenerate or not. Since the first six quadrics mentioned are ruled surfaces, the degenerate conics obtained are formed by a point, a double straight line, two parallel straight lines, or two intersecting straight lines; thus for the vector fields defined on these quadrics we get a bound for the maximum number of invariant straight lines contained in invariant planes of X. In the same way, if the conic is non-degenerate, it can be a parabola, an ellipse or a hyperbola and we provide a bound for the maximum number of invariant non-degenerate conics of the vector field X depending on each quadric Q and of the degrees m1, m2 and m3 of X.