  Superconductivity in topological insulators is expected to show very
unconventional features such as $p+ip$ order parameter, Majorana fermions etc.
However, the intrinsic superconductivity has been observed in a very limited
number of materials in which the pairing symmetry is still a matter of debate.
Here, we study the topological crystalline insulator
(Pb$_{1-x}$Sn$_x$)$_{1-y}$In$_y$Te, where a peculiar insulator to
superconductor transition was previously reported near the gap inversion
transition, when the system is nearly a 3-dimensional Dirac semimetal. Both the
existence of superconductivity near the 3-dimensional Dirac semimetal and the
occurrence of insulator to superconductor transition in an isotropic material
is highly unusual. We suggest that the observed phenomena are related to an
intrinsic instability of a 3-dimensional Dirac semimetal state in
(Pb$_{1-x}$Sn$_x$)$_{1-y}$In$_y$Te and "flattening" of the bulk valence and
conduction bands as they acquire a Mexican hat-like dispersion on the inverted
side of the phase diagram. This favors the pairing instability if the chemical
potential is pinned to these flat regions.
