the standard sphere.

The DPW method can produce spherical immersions of $\Sigma = {\mathbb{C}}\cup\left\{\infty \right\}$ with the potential $$\xi_{\mathbb{S}}(z,{\lambda}) = \begin{pmatrix}
0 & {\lambda}^{-1}dz \\
0 & 0
\end{pmatrix}$$ and initial condition $\Phi_{\mathbb{S}}(0,{\lambda}) = {\mathrm{I}}_2$. The potential is not regular at $z=\infty$ because it has a double pole there. However, the immersion will be regular at this point because $\xi_{\mathbb{S}}$ is gauge-equivalent to a regular potential at $z=\infty$. Indeed, consider on ${\mathbb{C}}^*$ the gauge $$G(z,{\lambda}) = \begin{pmatrix}
    z&0\\
    -{\lambda}& \frac{1}{z}
    \end{pmatrix}.$$ The gauged potential is then $$\xi_{\mathbb{S}}\cdot G (z,{\lambda}) = \begin{pmatrix}
    0 & {\lambda}^{-1}z^{-2}dz \\
    0&0
    \end{pmatrix}$$ which is regular at $z=\infty$. The holomorphic frame is $$\label{eqPhi_Ss}
\Phi_{\mathbb{S}}(z,{\lambda}) = \begin{pmatrix}
1 & {\lambda}^{-1}z \\
0 & 1
\end{pmatrix}$$ and its unitary factor is $$F_{\mathbb{S}}(z,{\lambda}) = \frac{1}{\sqrt{1+|z|^2}}\begin{pmatrix}
1 & {\lambda}^{-1}z\\
-{\lambda}{\overline{z}} & 1
\end{pmatrix}.$$ The induced CMC-$H$ immersion is $$f_{\mathbb{S}}(z) = \frac{1}{1+|z|^2}\begin{pmatrix}
1+ e^{2q}|z|^2 & 2z\sinh q \\
2{\overline{z}}\sinh q & 1+e^{-2q}|z|^2
\end{pmatrix}.$$ It is not easy to see that $f_{\mathbb{S}}(\Sigma)$ is a sphere because it is not centered at ${\mathrm{I}}_2$. To solve this problem, notice that $F_{\mathbb{S}}(z,e^{-q}) = R(q)\widetilde{F}_{\mathbb{S}}(z)R(q)^{-1}$ where $$\label{eqR(q)}
R(q) := \begin{pmatrix}
e^{\frac{q}{2}} & 0\\
0 & e^{\frac{-q}{2}}
\end{pmatrix}\in{\mathrm{SL}}(2,{\mathbb{C}})$$ and $$\widetilde{F}_{\mathbb{S}}(z) := \frac{1}{\sqrt{1+|z|^2}}\begin{pmatrix}
    1 & z\\
    -{\overline{z}} & 