We analyse, in the frame of the extended kinetic theory, travelling wa
ve solutions for the 2-velocity discrete models of the Boltzmann equat
ion. The extended formalism allows for absorption, creation, due to sc
attering with a background as well as for the existence of external so
urces of test particles. As a preliminary step, existence, uniqueness
and stability of equilibrium states are investigated. We find new, sol
ition-like solutions, and analyse their dependence on the physical par
ameters of the model.