On Darboux polynomials and rational first integrals of the generalized Lorenz system

Abstract The characterization of all Darboux polynomials and rational first integrals of the generalized Lorenz system, x ˙ = a ( y − x ) , y ˙ = b x + c y − x z , z ˙ = d z + x y , was published very recently in [K. Wu, X. Zhang, Bull. Sci. Math. 136 (2012) 291–308]. In this paper we improve that work in two aspects. On the one hand, we obtain the same results in a much more straightforward way. To do that, we show the equivalence under generic conditions, c ≠ 0 , between the Lorenz system and the generalized Lorenz system by means of a linear scaling in time and coordinates. Thus, from the well-known results on Darboux polynomials and its algebraic integrability for the Lorenz system, it is direct to obtain the corresponding results for the generalized Lorenz system. On the other hand, in the case c = 0 , we find a new Darboux polynomial of the generalized Lorenz system, not detected in the above paper, which is also a first integral. In this way, we complete the list provided by the authors of all Darboux polynomials and rational first integrals of the generalized Lorenz system.

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