  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.
