  In two-dimensional system with Rashba spin-orbit coupling, it is well-known
that superconductivity can have mixed spin-singlet and -triplet parity, and the
$\boldsymbol{d}$-vector of spin-triplet pairing is parallel to
$\boldsymbol{g}$-vector of Rashba spin-orbit coupling. Here, we propose a model
to describe a two-dimensional system with unconventional Rashba bands and study
its superconductivity. We show that the $\boldsymbol{d}$-vector of spin-triplet
pairing can be either parallel or perpendicular to $\boldsymbol{g}$-vector of
Rashba spin-orbit coupling depending on the different pairing interaction. We
also propose a junction to generate tunneling current depending on the
direction of $\boldsymbol{d}$-vector. It provides a detectable evidence to
distinguish these two different but very similar pairing channels. Furthermore,
we find this model can give arise to a subleading spin-singlet chiral $p$-wave
topological superconducting state. More significantly, we find that such
unconventional Rashba bands and unconventional superconudcting pairings can be
realized on surface of some superconducting topological materials, such as
trigonal layered PtBi$_{2}$.
