  Olver and Rosenau studied group-invariant solutions of (generally nonlinear)
partial differential equations through the imposition of a side condition. We
apply a similar idea to the special case of finite-dimensional Hamiltonian
systems, namely Hamilton-Jacobi, Helmholtz and time-independent Schr\"odinger
equations with potential on $N$-dimensional Riemannian and pseudo-Riemannian
manifolds, but with a linear side condition, where more structure is available.
We show that the requirement of $N-1$ commuting second-order symmetry
operators, modulo a second-order linear side condition corresponds to
nonregular separation of variables in an orthogonal coordinate system,
characterized by a generalized St\"ackel matrix. The coordinates and solutions
obtainable through true nonregular separation are distinct from those arising
through regular separation of variables. We develop the theory for these
systems and provide examples.
