  We consider optical media which feature antilinear symmetries. We show that:
(i) spectral singularities of such media (if any) are always self-dual, i.e.,
correspond to CPA-lasers; (ii) under the change of a system's parameter the
self-dual spectral singularity may split into a pair of isolated complex
conjugate eigenvalues, which corresponds to an unconventional and overlooked in
the most of previous studies scenario of the phase transition (known as
$PT$-symmetry breaking in systems obeying parity-time symmetry); (iii) if the
antilinear symmetry is local, i.e., does not involve any spatial reflection,
then no spectral singularity is possible. Our findings are illustrated with
several examples including a $PT$-symmetric bilayer and other complex
potentials discussed in recent literature.
