  This paper contains an analysis of rank-k solutions in terms of Riemann
invariants, obtained from interrelations between two concepts, that of the
symmetry reduction method and of the generalized method of characteristics for
first order quasilinear hyperbolic systems of PDEs in many dimensions. A
variant of the conditional symmetry method for obtaining this type of solutions
is proposed. A Lie module of vector fields, which are symmetries of an
overdetermined system defined by the initial system of equations and certain
first order differential constraints, is constructed. It is shown that this
overdetermined system admits rank-k solutions expressible in terms of Riemann
invariants. Finally, examples of applications of the proposed approach to the
fluid dynamics equations in (k+1) dimensions are discussed in detail. Several
new soliton-like solutions (among them kinks, bumps and multiple wave
solutions) have been obtained.
