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A typical problem with the conventional Galerkin approach for theconstruction of finite-mode models is to keep structural properties unaffectedin the process of discretization. We present two examples of finite-modeapproximations that in some respect preserve the geometric attributes inheritedfrom their continuous models: a three-component model of the barotropicvorticity equation known as Lorenz' maximum simplification equations [Tellus,\textbf{12}, 243--254 (1960)] and a six-component model of the two-dimensionalRayleigh--B\'{e}nard convection problem. It is reviewed that the Lorenz--1960model respects both the maximal set of admitted point symmetries and anextension of the noncanonical Hamiltonian form (Nambu form). In a similarfashion, it is proved that the famous Lorenz--1963 model violates thestructural properties of the Saltzman equations and hence cannot be consideredas the maximum simplification of the Rayleigh--B\'{e}nard convection problem.Using a six-component truncation, we show that it is again possible retainingboth symmetries and the Nambu representation in the course of discretization.The conservative part of this six-component reduction is related to theLagrange top equations. Dissipation is incorporated using a metric tensor.

Minimal atmospheric finite-mode models preserving symmetry and generalized Hamiltonian structures