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In this paper, we establish a relation between two seemingly unrelated
concepts for solving first-order hyperbolic quasilinear systems of partial
differential equations in many dimensions. These concepts are based on a
variant of the conditional symmetry method and on the generalized method of
characteristics. We present the outline of recent results on multiple Riemann
wave solutions of these systems. An auxiliary result concerning a modification
of the Frobenius theorem for integration is used. We apply this result in order
to show that the conditional symmetry method can deliver larger classes of
multiple Riemann wave solutions, through a simpler procedure, than the one
obtained from the generalized method of characteristics. We demonstrate that
solutions can be interpreted physically as a superposition of k single waves.
These theoretical considerations are illustrated by examples of
hydrodynamic-type systems in (n+1) dimensions.
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