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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.

In this letter we consider three nonhomogeneous deformations of Dispersive Water Wave (DWW) soliton equation and prove that their stationary flows are equivalent to three famous Painlev\'{e} equations, i.e. $P_{II}$, $P_{III}$ and $P_{IV},$ respectively.