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Agraïments: The second author is partially supported by FCT/Portugal through UID/MAT/04459/2013We consider the 5-dimensional Lorenz system \[ U' &= -V W b V Z, \\ V' &= UW-b UZ, \\ W'&= -U V,\\ X' &= -Z, \\ Z'&=b UV X \] where b \R \0\ and the derivative is with respect to T. This system describes coupled Rosby waves and gravity waves. First we prove that the number of functionally independent global analytic first integrals of this differential system is two. This solves an open question in the paper On the analytic integrability of the 5-dimensional Lorenz system for the gravity-wave activity, Proc. Amer. Math. Soc. 142 (2014), 531--537, where it was proved that this number was two or three. Moreover, we characterize all the invariant algebraic surfaces of the system, and additionally we show that it has only two functionally independent Darboux first integrals

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