$dH-\sqrt{H^2-K}d\varphi=0$, we have $$\sqrt{H(t)^2-K(t)}\,\varphi'(t)=H'(t).$$

Therefore, if there exists a primitive $\varphi(t)$ of the function $$\frac{H'(t)}{\sqrt{H(t)^2-K(t)}}$$ with $\sinh\varphi(t)/\sqrt{H(t)^2-K(t)}$ well-defined even at the umbilical points, then the Codazzi pair $(A,B)$, given as (\[AB\]), will exist on the whole surface $\Sigma$.

In addition, if $(I,II)$ is a special Weingarten pair satisfying $H=f(H^2-K)$, then we can parametrize in the way $H^2-K=t^2$ and $H=f(t^2)$. This allows us to take $$\label{varfi}
\varphi(t)=\int_0^t2f'(s^2)\,ds$$ whenever there exist umbilical points (i.e. $t=0$ has sense), or any primitive of $2f'(t^2)$ otherwise.

Thus, for special Weingarten pairs, the metric $A$ defined as in Corollary \[corolario3\] is always well-defined, because so is the function $\sinh\varphi(t)/t$.

The metric $A$ for special Weingarten surfaces in $\r^3$ and $\h^3$ was first defined by R.L. Bryant in [@Br]. In that work, he also found a holomorphic quadratic form for the metric $A$ which agrees with the Hopf differential of the pair $(A,B)$. Thanks to it, Bryant provided an easy proof of the fact that every topological sphere in $\r^3$ or $\h^3$ satisfying a special Weingarten relationship must be totally umbilical.

The abstract formulation which we have adopted in this work, allows us to extend this result to general special Weingarten surfaces.

Another remarkable fact is that, 