Complete integrability and analytical solutions of a Lorenz-like system

We present a complete analytical and numerical study of a generalized Lorenz-like system, characterizing its integrable structures and chaotic regimes. Through rigorous Painlev\'e analysis, we identify two distinct cases of complete integrability occurring at: (1)~$b=2 \sigma$ and (2)~$b=-r -3\sigma$. \ For each integrable case, we construct explicit first integrals ($I_1 = x^2 - 2\sigma z$ and $I_2 = (x+y)^2+\sigma z$, respectively) and derive families of exact solutions, including elliptic function solutions for the first case and exponential solutions for the second. The perturbed non-integrable system is investigated through Melnikov's method, revealing criteria for transverse homoclinic intersections and chaotic dynamics. Numerical bifurcation analysis demonstrates the transition from regular to chaotic behavior as perturbation strength increases, quantified by Lyapunov exponent calculations. Geometric singular perturbation theory identifies canard cycles in weakly dissipative regimes near critical parameter values. Our results provide a comprehensive framework for understanding both the integrable structure and the breakdown to chaos in this class of systems, with implications for controlling nonlinear dynamics in physical applications.