  The origin of spectral singularities in finite-gap singly periodic
PT-symmetric quantum systems is investigated. We show that they emerge from a
limit of band-edge states in a doubly periodic finite gap system when the
imaginary period tends to infinity. In this limit, the energy gaps are
contracted and disappear, every pair of band states of the same periodicity at
the edges of a gap coalesces and transforms into a singlet state in the
continuum. As a result, these spectral singularities turn out to be analogous
to those in the non-periodic systems, where they appear as zero-width
resonances. Under the change of topology from a non-compact into a compact one,
spectral singularities in the class of periodic systems we study are
transformed into exceptional points. The specific degeneration related to the
presence of finite number of spectral singularities and exceptional points is
shown to be coherently reflected by a hidden, bosonized nonlinear
supersymmetry.
