We provide necessary and sufficient conditions for both integrability and linearizability of a three dimensional vector field with quadratic nonlinearities. For our investigation we consider the case of (1 : −2 : 1)-resonance at the origin and in general non of the axes planes is invariant.Hence, we deal with a nine parametric family of quadratic systems. Some techniques like Darboux method are used to prove the sufficiency of the obtained conditions. For a particular three parametric subfamily we provide conditions to guarantee the non existence of a polynomial first integral.

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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 parameters for 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 completely 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.