quasi-periodicity of $\sigma$ via the relation [@abramowitz_handbook_1964 eq. 18.2.20] $$\begin{gathered}
\sigma\left(z+2M\omega+2N\omega^\prime\right) = \left(-1\right)^{M+N+MN}\sigma\left(z\right)\cdot \\
\mathrm{e}^{\left(z+M\omega+N\omega^\prime\right)\left[2M\zeta\left(\omega\right)+2N\zeta\left(\omega^\prime\right)\right]},\end{gathered}$$ with $M,N\in\mathbb{Z}$, we can then write for eq. $$\begin{gathered}
f\left(\tau + T\right) = A+B\tau+C_\xi\ln\frac{\sigma_\xi\left(\tau + a_\xi\right)}{\sigma_\xi\left(\tau + b_\xi\right)}
+C_\eta\ln\frac{\sigma_\eta\left(\tau + a_\eta\right)}{\sigma_\eta\left(\tau + b_\eta\right)}\\
+ BT + 2 m C_\xi\left(a_\xi - b_\xi \right)\zeta_\xi\left(\omega_{R,\xi}\right)\\
+ 2 n C_\eta\left(a_\eta - b_\eta \right)\zeta_\eta\left(\omega_{R,\eta}\right),\end{gathered}$$ or, more succinctly, $$f\left(\tau + T\right) = f\left(\tau\right) + D,$$ where $D$ is the constant $$\begin{gathered}
D = BT + 2 m C_\xi\left(a_\xi - b_\xi \right)\zeta_\xi\left(\omega_{R,\xi}\right)\\
+ 2 n C_\eta\left(a_\eta - b_\eta \right)\zeta_\eta\left(\omega_{R,\eta}\right).\end{gathered}$$ Thus, if $\xi\left( \tau \right)$ and $\eta\left( \tau \right)$ have commensurable periods, $\phi\left( \tau \right)$ is an arithmetic quasi-periodic function of $\tau$. The geometric meaning of this quasi-periodicity is that, after a quasi-period $T$, the test particle will be in a position that results from a rotation around the $z$ axis of the original position. The particle’s trajectory will thus draw a rotationally-symmetric figure in space.

Quasi-periodic orbits can be found via a numerical search for a set of initial conditions and constant acceleration field $\varepsilon$ that satisfies the commensurability relation on the periods of $\xi$ and $\eta$. The numerical search can be setup as the minimisation of the function $\left(m\omega_{R,\xi}-n\omega_{R,\eta}\right)^2$ for two 