- \frac{1}{4}\right| <\frac{1}{4} \quad \forall (t,{\lambda})\in(-T',T')\times {\mathbb{A}}_{{\rho}}$$ With this choice of $T'$, the function $\mu_t: {\mathbb{A}}_{{\rho}}\longrightarrow {\mathbb{C}}$ defined as the positive-real-part square root of $(-\det A_t)$ is holomorphic on ${\mathbb{A}}_{\rho}$ and real-valued on ${\mathbb{S}}^1$. Note that $\mu_t$ is also the positive-real-part eigenvalue of $A_t$ and thus $A_t=H_tD_tH_t^{-1}$ with $$\label{eqHtDt}
    H_t({\lambda}) = \frac{1}{\sqrt{2}}\begin{pmatrix}
    1 & \frac{-(r{\lambda}^{-1}+s)}{\mu_t({\lambda})}\\
    \frac{r{\lambda}+s}{\mu_t({\lambda})} & 1
    \end{pmatrix},\qquad D_t({\lambda}) = \begin{pmatrix}
    \mu_t({\lambda}) & 0\\
    0 & -\mu_t({\lambda})
    \end{pmatrix}.$$ Let $e^q<R<\rho$. For all $t\in(-T',T')$, $\mu_t\in\Lambda{\mathbb{C}}_{R}$ and the map $t\mapsto \mu_t$ is smooth on $(-T',T')$. Moreover, $H_t\in{\Lambda {\mathrm{SU}}(2)}_{R}$, $iD_t\in\Lambda{\mathfrak{su}}(2)_R$ and these functions are smooth with respect to $t$.

The bound $t<T'$ ensures that that $4 \det A_t({\lambda})$ is an integer only for $t=0$ and ${\lambda}=e^{\pm q}$. These points make the Fröbenius system resonant, but they are precisely the points that bear an extra piece of information due to the hypotheses on ${\mathcal{M}}(\Phi_t)(e^q)$ and $\Phi_0$. Allowing the parameter $t$ to leave the interval $(-T',T')$ would bring other resonance points and make Section \[sectionzAP\] invalid. This is why Theorem \[theoremPerturbedDelaunay\] does not state that the end of the immersion $f_t$ is a Delaunay end for all $t$.

At $t=0$, the change of basis $H_t$ in the diagonalisation of $A_t$ takes different values wether 