On symmetry propertiesof discrete velocity models of the Boltzmann equation

For a class of first order quasilinear PDEs containing all the variety of discrete velocity models (DVMs) the general form of the system of equations defining their Lie (point) symmetry groups is obtained. By means of these defining equations several classes of DVMs which admit only trivial symmetries (i.e. translations of the independent variables and dilatation) are found. It is also shown that besides trivial symmetries DVMs can admit only groups of translations of the dependent variables. It is established that the criterion of existence of these only possible non-trivial Lie symmetries of a spatial-inhomogeneous DVM is the incompleteness of the rank of a certain matrix with the elements being the cross-sections of the DVM. In conclusion several non-trivial Lie symmetries of the spatial-homogeneous Carleman model are presented.