Neglected constant curvature difference surfaces What are some surfaces where $ \kappa_1-\kappa_2$ is constant?  On a sphere where all are umbilical points.. is a special case.
For the $ \kappa_1+\kappa_2$ = constant case we have DeLaunay and Minimal surfaces.
$ \kappa_1,\kappa_2$ are the principal curvatures. 
EDIT1:
In the limited number of numerical integrations done so far as a surface of revolution I obtain a symmetrical U shaped meridian between vertical asymptote planes in one case and two cuspidal ring edges in another.

Another trivial case is a cylinder with $\kappa_{1}=\dfrac{1}{r}$ and $\kappa_{2}=0$.
How about another variation with $2\kappa_{1}-\kappa_{2}=0$?

Mylar balloon
  \begin{align*}
  \left(
     \begin{array}{c}
       x \\
       y \\
       z \\
     \end{array}
   \right)
  &= 
  r\left(
     \begin{array}{c}
       \text{cn} \left( u,\frac{1}{\sqrt{2}} \right) \cos v \\[5pt]
       \text{cn} \left( u,\frac{1}{\sqrt{2}} \right) \sin v \\[5pt]
       \sqrt{2} \left[
                  E \left(
                      \text{am} \left( u,\frac{1}{\sqrt{2}} \right) ,\frac{1}{\sqrt{2}}
                    \right)
                  -\frac{1}{2} u
                \right]
     \end{array}
   \right) \\
  \kappa_{1} &= \frac{1}{2r} \, \text{cn} \left( u,\frac{1}{\sqrt{2}} \right) \\
  \kappa_{2} &= \frac{1}{r} \, \text{cn} \left( u,\frac{1}{\sqrt{2}} \right)
\end{align*}

