  The versatile and exactly solvable Scarf II has been predicting, confirming
and demonstrating interesting phenomena in complex PT-symmetric sector, most
impressively. However, for the non-PT-symmetric sector it has gone
underutilized. Here, we present most simple analytic forms for the scattering
coefficients $(T(k),R(k),|\det S(k)|)$. On one hand, these forms demonstrate
earlier effects and confirm the recent ones. On the other hand they make new
predictions - all simply and analytically. We show the possibilities of both
self-dual and non-self-dual spectral singularities (NSDSS) in two non-PT
sectors (potentials). The former one is not accompanied by time-reversed
coherent perfect absorption (CPA) and gives rise to the parametrically
controlled splitting of SS in to a finite number of complex conjugate pairs of
eigenvalues (CCPEs). The latter ones (NSDSS) behave just oppositely: CPA but no
splitting of SS. We demonstrate a one-sided reflectionlessness without
invisibility. Most importantly, we bring out a surprising co-existence of both
real discrete spectrum and a single SS in a fixed potential. Nevertheless, the
complex Scarf II is not known to be pseudo-Hermitian ($\eta^{-1}
H\eta=H^\dagger$) under a metric of the type $\eta(x)$, so far.
