  We consider the role of degeneracy in Parity-Time (PT) symmetry breaking for
non-hermitian wave equations beyond one dimension. We show that if the spectrum
is degenerate in the absence of T-breaking, and T is broken in a generic manner
(without preserving other discrete symmetries), then the standard PT-symmetry
breaking transition does not occur, meaning that the spectrum is complex even
for infinitesimal strength of gain and loss. However the realness of the entire
spectrum can be preserved over a finite interval if additional discrete
symmetries X are imposed when T is broken, if X decouple all degenerate modes.
When this is true only for a subset of the degenerate spectrum, there can be a
partial PT transition in which this subset remains real over a finite interval
of T-breaking. If the spectrum has odd-degeneracy, a fraction of the degenerate
spectrum can remain in the symmetric phase even without imposing additional
discrete symmetries, and they are analogous to dark states in atomic physics.
These results are illustrated by the example of different T-breaking
perturbations of a uniform dielectric disk and sphere, and a group theoretical
analysis is given in the disk case. Finally, we show that multimode coupling is
capable of restoring the T-symmetric phase at finite T-breaking. We also
analyze these questions when the parity operator is replaced by another spatial
symmetry operator and find that the behavior can be qualitatively different.
