Summary: Semi-analytical investigations using the orientation-free discrete gas-kinetic theory are performed in order to link the non-equilibrium but stationary solitons found in the randomly and approximately boundary treatment for the interactions between gases and solid-surfaces with vorticity at the pseudo-flat walls. The stationary solitons thus obtained from the integrable system of partial differential equations are subjected to different but limited dissipations depending on the intrinsic parameter of rarefaction. The author also addresses the linear stability of our results in brief.