  We analyze the real-space paired state with the $\mathbf{k}$-dependent
superconducting gap in the presence of Rashba type spin-orbit coupling and
external magnetic field. We show that the $extended$ $s$-$wave$ pairing
symmetry is the most probable scenario to appear in the low-electron
concentration regime. According to our study, the van Hove singularity induced
by the spin-orbit coupling may lead to a significant enhancement of the
superconducting gap, critical temperature and critical magnetic field.
Moreover, the combined effect of the spin-orbit coupling and the external
magnetic field results in a non-zero total momentum of the Cooper pairs, which
is a characteristic feature of the so-called helical state. In such situation,
due to the C$_4$ symmetry breaking, a small $d$-$wave$ and $p$-$wave$
contributions to the pairing appear, which significantly change the character
of the helical state. The obtained results are discussed in the context of the
experimental data related with the unconventional superconducting features of
the transition metal oxide interfaces as well as the recently reported
supercurrent diode effect.

Superconductivity