The integrability conditions for two cubic vector fields (Q5926582)

The authors give necessary and sufficient conditions for the existence of a first integral of the form \(H(x,y)=xy+\sum_{k\geq 2} F_k(x,y),\) where \(F_k\) are homogeneous polynomials with complex coefficients, for the two complex 6-parameter families of cubic systems given by \[ \dot x=x+axy+bx^2+cy^2+dx^3,\quad \dot y=-y+Axy+By^2+Cx^2+Dy^3, \] where either \(a=A=0\) or \(b=B=0.\) The proof follows the usual steps. Firstly, they obtain some necessary conditions, which are given by polynomial relations among the coefficients. These relations are found by an algorithm developed by the first author in a previous paper [Differ. Equations 31, No.~6, 1023-1026 (1995); translation from Differ. Uravn. 31, No.~6, 1091-1093 (1995; Zbl 0861.34013)]. Secondly, they prove that these conditions are in fact sufficient. To this aim they use several methods: Darboux integrating factors, algebraic properties,\dots Maybe the difficult case is the one given by \(\dot x=x+x^2+dx^3,\) \(\dot x=-y-y^2+Cx^2,\) where it is proved the existence of a first integral of the form \(H(x,y)=\sum_{k\geq 1}H_k(y)x^k\) by using a method developed by \textit{A. Fronville, A. P. Sadovskii} and \textit{H. . Zołądek} [Fundam. Math. 157, No.~2-3, 191-207 (1998; Zbl 0943.34018)].