all $t$ and note that ${\mathcal{M}}(\Phi_0) = -{\mathrm{I}}_2$, which implies that, as a function of $t$, the derivative of ${\mathcal{M}}(\Phi_t)$ at $t=0$ is in ${\Lambda {\mathfrak{su}}(2)_\rho}$. On the other hand, Proposition 4 in Appendix B of [@raujouan] ensures that $$\frac{d{\mathcal{M}}(\Phi_t)}{dt}\mid_{t=0} = \left( \int_{\gamma} \Phi_0 \frac{d\xi_t}{dt}\mid_{t=0} \Phi_0^{-1} \right)\times {\mathcal{M}}(\Phi_0)$$ where $\gamma$ is a generator of $\pi_1(D_\epsilon^*,z_0)$. Expanding the right-hand side gives $$-\int_{\gamma} z^{A_0}\frac{d A_t}{dt}\mid_{t=0} z^{-A_0}z^{-1}dz - \int_{\gamma} z^{A_0}\frac{d C_t(z)}{dt}\mid_{t=0} z^{-A_0}dz \in{\Lambda {\mathfrak{su}}(2)_\rho}.$$ Using Proposition 4 of [@raujouan] once again, note that the first term is the derivative of ${\mathcal{M}}(z^{A_t})$ at $t=0$, which is in ${\Lambda {\mathfrak{su}}(2)_\rho}$ because ${\mathcal{M}}(z^{A_t})\in{\Lambda {\mathrm{SU}}(2)_\rho}$ and ${\mathcal{M}}(z^{A_0}) = -{\mathrm{I}}_2$. Therefore, the second term is also in ${\Lambda {\mathfrak{su}}(2)_\rho}$. Diagonalising $A_0$ with Proposition \[propDiagA\] and using $H_0\in\Lambda{\mathrm{SU}}(2)_{{R}}$ gives $$2i\pi{\mathop\mathrm{Res}}_{z=0} \left( z^{D_0}H_0^{-1}\frac{d}{dt}C_t(z)\mid_{t=0}H_0z^{-D_0} \right)\in \Lambda{\mathfrak{su}}(2)_{{R}}.$$ But using Equation , $$\begin{aligned}
    z^{D_0}H_0^{-1}\frac{d}{dt}C_t(z)\mid_{t=0}H_0z^{-D_0} &= z^{D_0}H_0^{-1}\left(\sum_{j=1}^{4}\widehat{c}_j(0)X_{0,j}\right)H_0z^{-D_0}\\
    &= \sum_{j=1}^{4}\widehat{c}_j(0)z^{D_0}E_jz^{-D_0} \\
    &= \begin{pmatrix}
    \widehat{c}_1(0) & z\widehat{c}_2(0)\\
    z^{-1}\widehat{c}_3(0) & \widehat{c}_4(0)
    \end{pmatrix}.
    \end{aligned}$$ Thus, $$2i\pi\begin{pmatrix}
        0 & 0\\
        \widehat{c}_3(0) & 0
        \end{pmatrix}\in\Lambda{\mathfrak{su}}(2)_{{R}}$$ which gives $\widehat{c}_3(0)=0$.

Let ${\lambda}_0\in\{e^q,e^{-q}\}$ and $t\neq 0$. Using the Fröbenius method (Theorem 4.11 of [@teschl] and Lemma 11.4 of [@taylor]) at the resonant point ${\lambda}_0$ ensures the existence of $\epsilon'>0$, $B,M\in{\mathcal{M}}(2,{\mathbb{C}})$ and a holomorphic map $P: D_{\epsilon'}\longrightarrow{\mathcal{M}}(2,{\mathbb{C}})$ such that for all $z\in D_{\epsilon'}^*$, $$\left\{
        