so-called helical basis to describe the normal state. We further derive the self-consistency relation in terms of the helical Green’s functions. Analysis of the self-consistency relation reveals that increasing the ratio $\alpha\,p_\text{F}/h$ leads to change the nature of superconducting correlations from interband to intraband. We also find an inhomogeneous superconducting phase in this regime, with a wave vector of the modulation scaling as $h_x^2\,h_z\,/\alpha$, for large values of $\alpha$. We finally provide a complete $(h_x,\, h_z,\, T)$-phase diagram for different orientations of the Zeeman field and several values of the spin-orbit coupling constant $\alpha$.

\[sec:weak\_soc\] Weak spin-orbit coupling
==========================================

We consider a single 1D superconducting wire deposited on top of a substrate (Fig. \[schema\_system\]). The mirror symmetry breaking induces Rashba spin-orbit interaction in the wire. A magnetic field can be applied in the plane $x-z$ corresponding to the surface of the substrate, thereby generating the Zeeman coupling. After introducing the Hamiltonian, we derive the corresponding exact Gor’kov equations for the Green’s functions of the 1D wire. The self-consistency relation is then studied in the limit of weak spin-orbit interaction and small wave-vector $q$ of the modulated superconducting order parameter.

\[subsec:model\_weak\] Model and Hamiltonian
--------------------------------------------

Let us consider a superconducting wire 