Lie symmetries of the Lorenz model

We study the generalized symmetries of the Lorenz model to find the parameter values at which one or more time-dependent integrals of motion exist. In these cases the integrals are found trivially from the symmetries themselves. A complete study of the one completely algebraically integrable case shows: (a) the dynamics can be integrated exactly, by reducing it first to a lower-dimensional system; (b) the symmetry vector field is Hamiltonian. These facts hold for other dissipative, completely integrable dynamical systems as well. The analytic study of a natural two-form reveals that it is an entire function of time. The foliation of phase space induced by the two-form for the partially integrable cases has a simple description in terms of the coefficients occurring in the Laurent series expansions of the dependent variables.