Darboux integrability of the simple chaotic flow with a line equilibria differential system

Abstract In this paper, we study the first integrals of Darboux type of the differential system x ˙ = y , y ˙ = − x + y z , z ˙ = − x − a x y − b x z , which exhibits chaotic phenomena for suitable chosen values of the real parameters a and b. We show that the system has no polynomial, rational, or Darboux first integrals for any value of a and b. All the Darboux polynomials of the system are derived together with its exponential factors. This study effectively exploits the benefits inherent in weight-homogenous polynomials to solve linear partial differential equations.