Indeed, when the spin-orbit and Zeeman fields are parallel, the SOC can be gauged out, and we obtain an effective Hamiltonian without SOC.

In the opposite case, namely when the Zeeman field is transverse to the SOC ($h_z = 0$), we obtain a rather different effect: The wave-vector $q$ vanishes, and $T_c$ is not modified with respect to $T_{c0}$ within leading order in $h_x/\alpha\,p_\text{F} \ll 1$. As we mentioned previously, this result comes from the fact that we neglect interband correlations in the self-consistency relation Eq. (\[app:SCE\_fin\]).

We can also compare the $(h_x,\,h_z,\,T)$-phase diagrams for different values of the spin-orbit coupling constant $\alpha$ for $h_x = h_z$, as illustrated in Fig. \[diag\_phaseSO\]. The blue curve represents the transition line between the normal and superconducting states in the absence of SOC, whereas the three others correspond to three finite values of $\alpha$. Globally, the presence of spin-orbit interaction stabilizes the superconducting phase in a broader range of fields and temperature. Let us focus on the transition lines corresponding to $\alpha \neq 0$. At temperatures $T > T^\star = 0.56\,T_{c0}$, these curves cannot be distinguished from each other due to the very small values of $q$. These ones can be estimated from 