  We consider solutions of two-dimensional $m \times m$ systems hyperbolic
conservation laws that are constant in time and along rays starting at the
origin. The solutions are assumed to be small $L^\infty$ perturbations of a
constant state and entropy admissible, and the system is assumed to be
non-strictly hyperbolic with eigenvalues of constant multiplicity. We show that
such a solution, initially assumed bounded, must be a special function of
bounded variation, and we determine the possible configuration of waves. As a
corollary, we extend some regularity and uniqueness results for some
one-dimensional Riemann problems.
