  Recent experiments show the spontaneous breaking of rotational symmetry in
the superconducting topological insulators M$_{x}$Bi$_{2}$Se$_{3}$ (M
represents Cu, Sr, or Nd), suggesting that the pairing belongs to a
two-dimensional representation of the $D_{3d}$ symmetry group of the crystal.
Motivated by these progresses, we construct an exhaustive list of possible
two-component pairings of the M$_{x}$Bi$_{2}$Se$_{3}$ superconductors, both for
the odd-parity $E_{u}$ representation and for the even-parity $E_{g}$
representation. Starting from a tight-binding model for the normal phase of
Bi$_{2}$Se$_{3}$ and M$_{x}$Bi$_{2}$Se$_{3}$, we firstly construct the pairing
channels in the spin-orbital basis, up to second-nearest-neighbor pairing
correlations in the basal plane. We then infer the properties of these pairings
by transforming them to the band (pseudospin) basis for the conduction band. A
comparison with the key experimental consensuses on M$_{x}$Bi$_{2}$Se$_{3}$
superconductors shows that the true pairings should also be multichannel.
Besides a nematic and time-reversal symmetric pairing combination, the other
pairings that we have identified are chiral and nematic at the same time, which
may be nonunitary and have a spontaneous magnetization. A complementary set of
experiments are proposed to identify the true pairing symmetries of these
superconductors and their evolution with the doping concentration $x$.
