Analytical Solutions of the Lorenz System

The Lorenz system is considered. The Painlevé test for the third-order equation corresponding to the Lorenz model at $\sigma \ne 0$ is presented. The integrable cases of the Lorenz system and the first integrals for the Lorenz system are discussed. The main result of the work is the classification of the elliptic solutions expressed via the Weierstrass function. It is shown that most of the elliptic solutions are degenerated and expressed via the trigonometric functions. However, two solutions of the Lorenz system can be expressed via the elliptic functions.