Polynomial first integrals of quadratic vector fields
The goal of this paper is to characterize those quadratic differential equations in the plane having a polynomial first integral, and to provide an explicit expression of these systems as well as of their polynomial first integrals. It is well known that if the system writes as \[ \dot x=P(x,y), \quad \dot y=Q(x,y) \] it is not restrictive to assume that \(P(x,y)\in\{1+xy, xy, y+x^2, y, 1+x^2, -1+x^2, x^2, x, 1, 0\}\) and \(Q(x,y)=a+bx+cy+dx^2+exy+fy^2.\) The authors consider only the cases where the system has finitely many critical points and solve completely the problem. For instance, their result when \(P(x,y)=1+x^2\) says that in this case the system has a polynomial first integral \(H(x,y)\) if and only if, either \(f=c=b=d=0,\) \(e=-2k\) with \(k\) a positive integer and \(a=1,\) and then \[ H(x,y)=y(1+x^2)^k-\sum_{j=0}^{k-1}{{k-1}\choose{j}}{{x^{2j+1}}\over{2j+1}}, \] or \(f=a=c=b=d=0\) and \(e=-p/q\) being a negative rational number, and then \(H(x,y)=(1+x^2)^py^{2q}.\) The more complicated cases in their study are the more generic ones, \textit{i.e.} \(P(x,y)=1+xy\) and \(P(x,y)=xy.\)
scientific article; zbMATH DE number 5083413