  In this paper, we describe certain crucial steps in the development of an
algorithm for finding the Riemann solution in systems of conservation laws. We
relax the classical hypotheses of strict hyperbolicity and genuine nonlinearity
of Lax. First, we present a procedure for continuing wave curves beyond points
where characteristic speeds coincide, i.e., at wave curve points of maximal
codimensionality. This procedure requires strict hyperbolicity on both sides of
the coincidence locus. Loss of strict hyperbolicity is regularized a
Generalized Jordan Chain, which serves to construct a four-fold submanifold
structure on which wave curves can be continued. Second, we analyze the case of
loss of genuine nonlinearity. We prove a new result: the existence of composite
wave curves when the composite wave traverses either the inflection locus or an
anomalous part of the non-local composite wave curve. In this sense, we find
conditions under which the composite field is well defined and its
singularities can be removed, allowing use of our continuation method. Finally,
we present numerical examples for a non-strictly hyperbolic system of
conservation laws.
