Motivated by observations of saturation overshoot this article investigatesgeneric classes of smooth travelling wave solutions of a system of two-couplednonlinear parabolic partial differential equations resulting from a flux function ofhigh symmetry. All boundary value problems of the travelling wave Ansatz, thatlead to smooth travelling wave solutions, are systematically explored. A complete, visual and computationally useful representation of the five dimensional manifold connecting wave velocities and boundary data is found by using a dynamical system approach. The travelling waves exhibit monotonic, non-monotonic or plateau-shaped behaviour. Special attention is given to the non-monotonic proles. The stability of the travelling waves is studied by numerically solving the full system of the partial dierential equations with an ecient and accurate adaptive movinggrid solver.