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We consider the role of degeneracy in parity-time () symmetry breaking for non-Hermitian wave equations beyond one dimension. We show that if the spectrum is degenerate in the absence of breaking, and is broken in a generic manner (without preserving other discrete symmetries), then the standard 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 are imposed when is broken, if decouples all degenerate modes. When the decoupling holds only for a subset of the degenerate spectrum, there can be a partial transition in which this subset remains real over a finite interval of breaking. If the spectrum has odd degeneracy, a fraction of the degenerate spectrum can remain in the symmetric phase even …