= \frac{1}{\sqrt{2}}\begin{pmatrix}
    1 & -2\left(re^{-q}+s\right)\\
    2\left(re^q+s\right) & 1
    \end{pmatrix}, \qquad D = \begin{pmatrix}
    \frac{1}{2}& 0\\0 & \frac{-1}{2}
    \end{pmatrix}.$$ Noting that $\exp\left(i\theta D\right)$ acts as a rotation of angle $\theta$ around the axis ${\mathrm{geod}}({\mathrm{I}}_2,\sigma_3)$ and that $H_{r,s}$ acts as an isometry of ${\mathbb{H}}^3$ independant of $\theta$ shows that $\exp\left(i\theta A_{r,s}(e^{-q})\right)$ acts as a rotation around the axis $H_{r,s}\cdot {\mathrm{geod}}({\mathrm{I}}_2,\sigma_3)$ and that $f_{r,s}$ is CMC $H>1$ immersion of revolution of ${\mathbb{C}}^*$ into ${\mathbb{H}}^3$ and by definition (as in [@meeks]) a Delaunay immersion.

#### The weight as a parameter.

For a fixed $H>1$, CMC $H$ Delaunay surfaces in ${\mathbb{H}}^3$ form a family parametrised by the weight. This weight can be computed in the DPW framework: given a solution $(r,s)$ of , the weight $w$ of the Delaunay surface induced by the DPW data $({\mathbb{C}}^*,\xi_{r,s},1,{\mathrm{I}}_2)$ reads $$\label{eqPoidsDelaunayPure}
w = 2\pi \times 4 rs\sinh q$$ (see [@loopgroups] for details).

Writing $t:=\frac{w}{2\pi}$ and assuming $t\neq 0$, Equations and imply that $$\label{eqSystemers}
        \left\{
        \begin{array}{l}
        t\leq T_1,\\
        r^2= \frac{1}{8}\left( 1-2Ht \pm 2 \sqrt{T_1-t}\sqrt{T_2-t} \right),\\
        s^2 = \frac{1}{8}\left( 1-2Ht \pm 2\sqrt{T_1-t}\sqrt{T_2-t} \right)
        \end{array}
        \right.$$ with $$T_1 = \frac{\tanh\frac{q}{2}}{2}<\frac{1}{2\tanh\frac{q}{2}}=T_2.$$

First, note that and imply $$r^2 + s^2 =\frac{1}{4}\left( 1 - 2t\coth q \right) = \frac{1}{4}(1-2Ht)$$ and thus $$\label{eqt<T2}
        t\leq \frac{H}{2}<T_2.$$ 