pair on a surface ${\Sigma}$ with mean and extrinsic curvatures $H$ and $K$ respectively. Let $\varphi$ be a differentiable function on ${\Sigma}$ such that the function $\sinh\varphi/\sqrt{H^2-K}$ can be differentiably extended to ${\Sigma}$. Then the fundamental pair $$\label{AB}
\begin{array}
{l} 
{\displaystyle A=\cosh\varphi\,I+\frac{\sinh\varphi}{\sqrt{H^2-K}}\,II'}\\[6mm]
{\displaystyle B=\sqrt{H^2-K}\sinh\varphi\,I+\cosh\varphi\,II'},
\end{array}$$ has mean curvature $H(A,B)=0$ and extrinsic curvature $K(A,B)=-(H^2-K)$. In addition, the following conditions are equivalent:

-   $(A,B)$ is a Codazzi pair,

-   the Hopf differential of $(A,B)$ is holomorphic for the conformal structure induced by $A$,

-   $
    dH-\sqrt{H^2-K}d\varphi=0.$

Next we see some situations where the corollary above can be used. In order to do that and following the classical notation, we give the following definition

\[sw\] We say that a Codazzi pair $(I,II)$ is a special Weingarten pair if there exists a differentiable function $f$ defined on an interval ${\cal
J}\subseteq [0,\infty)$ such that its mean curvature $H$ and extrinsic curvature $K$ satisfy $$H=f(H^2-K).$$

Now, let us suppose that the mean and extrinsic curvatures of a Codazzi pair $(I,II)$ satisfy a general Weingarten relationship $W(H,K)=0$. Let us parametrize by taking $H=H(t)$, $K=K(t)$, for $t$ varying in a certain interval. Then, if we look for a solution of the type $\varphi=\varphi(t)$ for the previous Equation 