We investigate discrete two-particle collision models on infinite grids which are invariant under integer shifts. Such models can be generated by constructing so-called single-layer systems which are based on one single type of collision. The work discusses fundamental features which are inherent to all such systems and their application as numerical simulation tools for the Boltzmann collision operator. One focus is the grid deficiencies prohibiting full translational and rotational invariance. The paper restricts to 2D velocity models. However, the results can be extended to 3D.