Riemann’s initial value problem has been studied in our paper [9] for a rather broad class of such hyperbolic systems with which rotational symmetry creates a specific kind of degeneracy: under natural genericity assumptions it is found to have a unique stable centered solution. This result, which holds locally near degenerate points in state space, applies to important examples from continuum mechanics. In sec.3 of the present report we will see that in all these cases the pattern of the degenerate transverse waves is isomorphic to that of a rather simple standard model. In sec.4 important qualitative features of the wave pattern in the general case are observed by means of simple explicit calculations on the model. Seel introduces to the situation by reviewing previous results for reference and sec.2 discusses the concept of stability that we use.