First integrals of the Maxwell–Bloch system

We investigate the analytic, rational and $C^1$ first integrals of the Maxwell–Bloch system \begin{equation*} \dot{E}=-\kappa E+gP,\quad \dot{P}=-\gamma _{\bot }P+gE\triangle , \quad \dot{\triangle }=-\gamma _{\Vert }(\triangle -\triangle _0)-4gPE, \end{equation*} where $\kappa , \gamma _{\bot }, g, \gamma _{\Vert }, \triangle _0$ are real parameters. In addition, we prove this system is rationally non-integrable in the sense of Bogoyavlenskij for almost all parameter values.