$P^1\in {\mathcal{C}}^1((-T',T'),{{\Lambda {\mathfrak{sl}}(2,{\mathbb{C}})}_R})$.

The $z^AP$ form of $\Phi_t$ {#sectionzAP}
---------------------------

The map $P^1$ defined above allows us to use the Fröbenius method in a loop group framework and in the non-resonant case, that is, for all $t$ (Proposition \[propzAP1\]). The techniques used in [@raujouan] will then apply in order to gauge the $Mz^AP$ form and gain an order on $z$ (Proposition \[propzAP2\]).

\[propzAP1\] There exists $\epsilon'>0$ such that for all $t\in(-T',T')$ there exist $M_t\in\Lambda{\mathrm{SL}}(2,{\mathbb{C}})_{R}$ and a holomorphic map $P_t: D_{\epsilon'}\longrightarrow {\Lambda {\mathrm{SL}}(2,{\mathbb{C}})_R}$ such that for all $z\in D_{\epsilon'}^*$, $$\Phi_t(z) = M_tz^{A_t}P_t(z).$$ Moreover, $M_t$ is ${\mathcal{C}}^1$ with respect to $t$, $M_0={\mathrm{I}}_2$, $P_t(z)$ is ${\mathcal{C}}^1$ with respect to $(t,z)$, $P_0(z) = {\mathrm{I}}_2$ for all $z$ and $P_t(0)={\mathrm{I}}_2$ for all $t$.

For all $k\in {\mathbb{N}}^*$ and $t\in(-T',T')$, define the linear map $${
    \begin{array}{ccccc}
        {\mathcal{L}}_{t,k} & : & \Lambda{\mathcal{M}}(2,{\mathbb{C}})_{{\rho}} & \longrightarrow & \Lambda{\mathcal{M}}(2,{\mathbb{C}})_{{\rho}} \\
        & & X & \longmapsto & [A_t,X] + kX. \\
    \end{array}
}$$ Use the bases ${\mathcal{B}}_t$ and restrict ${\mathcal{L}}_{t,k}$ to $\Lambda{\mathcal{M}}(2,{\mathbb{C}})_R$ to get $$\label{eqDiagLlk}
    {\mathrm{Mat}}_{{\mathcal{B}}_t}{\mathcal{L}}_{t,k} = \begin{pmatrix}
    k & 0 &0&0\\
    0&k+2\mu_t&0&0\\
    0&0&k-2\mu_t&0\\
    0&0&0&k
    \end{pmatrix}.$$ Note that $$\det {\mathcal{L}}_{t,k} = k^2(k^2-4\mu_t^2).$$ Thus, for all $k\geq 2$, $\det {\mathcal{L}}_{t,k}$ is an invertible element of $\Lambda{\mathbb{C}}_{{R}}$, which implies that ${\mathcal{L}}_{t,k}$ is invertible 