The reducibility of partially invariant solutions of systems of partial differential equations

Ovsiannikov's partially invariant solutions of differential equations generalize Lie's group invariant solutions. A partially invariant solution is only interesting if it cannot be discovered more readily as an invariant solution. Roughly, a partially invariant solution that can be discovered more directly by Lie's method is said to be reducible. In this paper, I develop conditions under which a partially invariant solution or a class of such solutions must be reducible, and use these conditions both to obtain non-reducible solutions to a system of hyperbolic conservation laws, and to demonstrate that some systems have no non-reducible solutions. I also demonstrate that certain elliptic systems have no non-reducible solutions.