Remarks on the integrabililty of the Lorenz System

In this work, we study the integrability, as well as the dynamics of the Lorenz System. This include a very useful identity:$$\beta z^2(\sigma t)+y^2(\beta \sigma t)=\rho x^2(\beta t)+\nu e^{-2\beta \sigma t},$$where $\nu \in \mathbb{R}$ is a constant. And we will see some applications of this identity.