  Two new approaches to solving first-order quasilinear elliptic systems of
PDEs in many dimensions are proposed. The first method is based on an analysis
of multimode solutions expressible in terms of Riemann invariants, based on
links between two techniques, that of the symmetry reduction method and of the
generalized method of characteristics. A variant of the conditional symmetry
method for constructing this type of solution is proposed. A specific feature
of that approach is an algebraic-geometric point of view, which allows the
introduction of specific first-order side conditions consistent with the
original system of PDEs, leading to a generalization of the Riemann invariant
method for solving elliptic homogeneous systems of PDEs. A further
generalization of the Riemann invariants method to the case of inhomogeneous
systems, based on the introduction of specific rotation matrices, enables us to
weaken the integrability condition. It allows us to establish a connection
between the structure of the set of integral elements and the possibility of
constructing specific classes of simple mode solutions. These theoretical
considerations are illustrated by the examples of an ideal plastic flow in its
elliptic region and a system describing a nonlinear interaction of waves and
particles. Several new classes of solutions are obtained in explicit form,
including the general integral for the latter system of equations.
