On the analytic integrability of the $5$–dimensional Lorenz system for the gravity–wave activity

Abstract:

For the $5$-dimensional Lorenz system \begin{eqnarray*} dU/dT &=& -VW+b VZ,\\ dV/dT &=& UW-b UZ,\\ dW/dT &=& -UV, \\ dX/dT &=& -Z,\\ dZ/dT &=& b UV+X \end{eqnarray*} (with $b\in \mathbb {R}$ a parameter), describing coupled Rosby and gravity waves, we prove that it has at most three functionally independent global analytic first integrals and exactly three functionally independent global analytic first integrals when $b=0$. In this last case the system is completely integrable with an additional functionally independent first integral which is not globally analytic.