  We study a class of qubit non-Markovian general Pauli dynamical maps with
multiple singularities in the generator. We discuss a few easy examples
involving trigonometric or other non-monotonic time dependence of the map, and
discuss in detail the structure of channels which don't have any trigonometric
functional dependence. We demystify the concept of a singularity here, showing
that it corresponds to a point where the dynamics can be regular but the map is
momentarily non-invertible, and this gives a basic guideline to construct such
non-invertible non-Markovian channels. Most members of the channels in the
considered family are quasi-eternally non-Markovian (QENM), which is a broader
class of non-Markovian channels than the eternal non-Markovian channels. In
specific, the measure of quasi-eternal non-Markovian (QENM) channels in the
considered class is shown to be $\frac{2}{3}$ in the isotropic case, and about
0.96 in the anisotropic case.
