even in momentum terms will remain, allowing to transform the integral over $p$ into an integral over $\xi$ following: $$\int_{-\infty}^{+\infty} \mathrm{d}p =  2\,\int_{-\mu}^{+\infty} N\left(\xi\right) \mathrm{d}\xi  \;,
\label{eq:change_int}$$ where the density of states at energy $\xi$ reads : $$N\left(\xi\right) = \displaystyle\sqrt{\frac{m}{2\left(\xi + \mu\right)}} \, .$$ For a finite spin-orbit coupling, the Green’s function $F^\dagger_{\downarrow\uparrow}$ contains terms proportional to $p^2 = 2\,m\left(\xi + \mu\right)$ and $p^4$ in the numerator. Then, the presence of the small parameters $\xi/\mu$ and $\xi^2/\mu^2$ in $G$ implies that we should expand $N\left(\xi\right)$ at the second order in $\xi/\mu$. This is a specificity of spin-coupled systems. In the absence of SOC, it is sufficient to keep the density of state $N\left(\xi\right)$ at zero order in $\xi/\mu$ to compute the Green functions. Finally, the integral Eq. ($\ref{eq:change_int}$) becomes: $$\int_{-\infty}^{+\infty} \mathrm{d}p \rightarrow \frac{2}{v_\text{F}} \int_{-\infty}^{+\infty} \left(1 - \frac{\xi}{2\,\mu} + \frac{3\,\xi^2}{8\,\mu^2}\right) \mathrm{d}\xi  \;,
\label{eq:int_xi}$$ where $v_\text{F} = \displaystyle\sqrt{2\,m/\mu}$. The $\xi$ integration is then performed using the residue technique.

The main result is the presence of a linear term in $q$ which leads to the emergence of a modulated superconducting state at wave-vector $q_0$ given by : $$\hbar q_0   = \frac{127\,m^2\,h_x^2\,h_z\,\alpha^3}{8\,\pi^4\,T_c^4}\,\frac{\zeta(7)}{\zeta(3)} \;,
    \label{q0_weak}$$ where the factor $\hbar$ has been restored (previously $\hbar=1$ in 