Darboux and Analytic First Integrals of the Generalized Michelson System

The purpose of this work is to demonstrate that, for any value of a_1, a_2, and a_3, the generalized Michelson system u = v,v =w,w = a_1 +a_2 v+ a_3 w-u^2/2 has neither a Darboux nor a rational first integral. Furthermore, we shall demonstrate that for a_3<0,(2 a_1 ) >0 and a_3 a_2-(2 a_1 ) >0, this system has no global C^1 first integrals. Additionally, the analytic first integral of this system for a generic condition is investigated near the equilibrium point ((2 a_1 ),0,0). The purpose of this work is to demonstrate that, for any value of a_1, a_2, and a_3, the generalized Michelson system u = v,v =w,w = a_1 +a_2 v+ a_3 w-u^2/2 has neither a Darboux nor a rational first integral. Furthermore, we shall demonstrate that for a_3<0,(2 a_1 ) >0 and a_3 a_2-(2 a_1 ) >0, this system has no global C^1 first integrals. Additionally, the analytic first integral of this system for a generic condition is investigated near the equilibrium point ((2 a_1 ),0,0).