0 & r{\lambda}^{-1}+s\\
    r{\lambda}+s & 0
    \end{pmatrix}, \qquad r,s\in{\mathbb{R}}, \quad {\lambda}\in{\mathbb{S}}^1,$$ and initial condition $\Phi_{r,s}(1) = {\mathrm{I}}_2$. With these data, the holomorphic frame reads $$\Phi_{r,s}(z) = z^{A_{r,s}}.$$ The unitary frame $F_{r,s}$ is not explicit, but the DPW method states that the map $f_{r,s}={ \mathrm{Sym}}_q(F_{r,s})$ is a CMC $H$ immersion from the universal cover $\widetilde{{\mathbb{C}}^*}$ of ${\mathbb{C}}^*$ into ${\mathbb{H}}^3$.

#### Monodromy.

Computing the monodromy along $\gamma(\theta) = e^{i\theta}$ for $\theta\in [0,2\pi]$ gives $${\mathcal{M}}\left(\Phi_{r,s}\right):= {\mathcal{M}}_\gamma\left(\Phi_{r,s}\right) = \exp\left(2i\pi A_{r,s}\right).$$ Recall that $r,s\in{\mathbb{R}}$ to see that $iA_{r,s}\in{\Lambda {\mathfrak{su}}(2)_\rho}$, and thus ${\mathcal{M}}\left(\Phi_{r,s}\right)\in{\Lambda {\mathrm{SU}}(2)_\rho}$: the first equation of is satisfied. To solve the second one, we look forward to finding $r$ and $s$ such that $A_{r,s}(e^{-q})^2=\frac{1}{4}{\mathrm{I}}_2$, which will imply that ${\mathcal{M}}\left(\Phi_{r,s}\right)(e^{-q})=-{\mathrm{I}}_2$. This condition is equivalent to $$\label{eqrs}
    r^2+s^2+2rs\cosh q = \frac{1}{4}.$$ Seeing this equation as a polynomial in $r$ and computing its discriminant ($1+4s^2\sinh^2q>0$) ensures the existence of an infinite number of solutions: given a couple $(r,s)\in{\mathbb{R}}^2$ solution to , $f_{r,s}$ is a well-defined CMC $H$ immersion from ${\mathbb{C}}^*$ into ${\mathbb{H}}^3$.

#### Surface of revolution.

Let $(r,s)\in{\mathbb{R}}^2$ satisfying and let $\theta\in{\mathbb{R}}$. Then, $$\Phi_{r,s}\left(e^{i\theta}z\right) = \exp\left(i\theta A_{r,s}\right)\Phi_{r,s}(z).$$ Using $iA_{r,s}\in{\Lambda {\mathfrak{su}}(2)_\rho}$ and diagonalising $A_{r,s}(e^{-q})$ gives $$\begin{aligned}
    f_{r,s}\left(e^{i\theta} z\right) &= \exp\left(i\theta A_{r,s}(e^{-q})\right)\cdot f_{r,s}(z)\\
    &=\left( H_{r,s}\exp\left(i\theta D\right)H_{r,s}^{-1}\right)\cdot f_{r,s}(z)\end{aligned}$$ where $$H_{r,s} 