Integrability and dynamics Analysis of the Chaos Laser System

In this article the complex dynamics of a laser model, which externally injected class 𝐵 which is described by a system of three nonlinear ordinary differential equations with two parametersfor field intensity phase and population inversion, are studied. In particular, we investigate the integrability and nonintegrabilty of laser system in three dimension. We prove that system is complete integrable only when the parameters are zero. Particularly, we study polynomial, rational, Darboux and analytic first integrals of the mentioned system. Moreover, we compute all the invariant algebraic surfaces and exponential factors of this system. We find sufficient conditions for the existence of periodic orbits emanating from an equilibrium point origin of a laser differential system with a first integral.