in our abstract context, we are able to recover every special Weingarten pair as follows

\[corolario4\] Let ${\Sigma}$ be a surface and $f$ a differentiable function defined on an interval ${\cal J}\subseteq [0,\infty)$. Let us take a primitive $\varphi(t)$ of $2f'(t^2)$ on that interval such that the function $\sinh\varphi(t)/t$ is well-defined. Then every special Weingarten pair $(I,II)$ on ${\Sigma}$ satisfying $H=f(H^2-K)$ is given by $$\label{III}
\begin{array}
{l}
{\displaystyle I=-\frac{\sinh\varphi(t)}{t}\,Q+\cosh\varphi(t)\,A-
\frac{\sinh\varphi(t)}{t}\,\overline{Q},}\\[5mm]
{\displaystyle
II-f(t^2)\,I=-\cosh\varphi(t)\,Q+t\,\sinh\varphi(t)\,A-\cosh\varphi(t)\,\overline{Q},
}
\end{array}$$ where $A$ is a Riemannian metric on ${\Sigma}$ and $Q$ a holomorphic 2-form for $A$ such that the image of the function $t:{\Sigma}\fl[0,\infty)$ defined as $2\,|Q|=t\,A$ is contained in ${\cal
J}$. In particular, $t^2=H^2-K$.

It suffices to observe that given a special Weingarten pair $(I,II)$, if we take $H^2-K=t^2$ (and therefore $H=f(t^2)$), we have already proved that there exists $\varphi=\varphi(t)$, primitive of $2f'(t^2)$, in the conditions of Corollary \[corolario3\]. Thus, there exists a pair $(A,B)$ made up of a Riemannian metric $A$ and a quadratic form $B$ which can be written as $B=Q+\overline{Q}$, since $H(A,B)=0$. Besides, the Hopf differential $Q$ of $(A,B)$ is a holomorphic 2-form for the metric $A$ and $$t^2\,=\,H^2-K\,=\,-K(A,B)\,=\,4\,\frac{|Q|^2}{|A|^2}.$$

Summing up, using (\[AB\]) it is possible to recover $(I,II)$ from $(A,B)$ as (\[III\]). 