..a small addition to Professor Bryant's answer
For this class of surfaces, where $k1-k2 = constant$, we have introduced the name "Costant Skew Curvature Surfaces" (CSkC-surfaces) and we have studied an aspect concerning the Bonnet-surfaces.
It is well know that the famous question that Bonnet asked was: "When does there exist an isometric embedding $x:M^2 \rightarrow R^3$ such that the mean curvature function of the immersion is $H$?"
Our work was born from the question: Can a surface be CSkC and Bonnet at the same time, and, if that is the case, what does it represent?
We showed that the CSkC-surfaces with principal curvatures ($k_1$ and $k_2$) nonconstant cannot contain any Bonnet-surfaces, so if and only if $k_1$ and $k_2$ are both constant the class of CSkC-surfaces can admit Bonnet-surfaces.
This means that the only CSkC-surfaces for which exists a nontrivial isometric deformation preserving the mean curvature $H$ are (patch of) a circular cylinders.
see: https://projecteuclid.org/euclid.jgsp/1518577293
Another aspect of the CSkC-surfaces is that if we setting $k_1-k_2=constant$ renders the shape equation for an elastic membrane equivalent to the Schrodinger equation for a particle on the same surface.. then the same equations have the same solutions...
see: https://iopscience.iop.org/article/10.1088/0143-0807/38/1/015405/pdf