..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 (Alexander Pigazzini, Magdalena Toda): 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 (Victor Atanasov, Rossen Dandoloff): https://iopscience.iop.org/article/10.1088/0143-0807/38/1/015405/pdf
By relating the two works we can for example say that:
"if you want to look for an isometric deformation that preserve this relation (shape equation equivalent to the Schrodinger equation) then you can get the result only if the surface is (patch of) circular cylinder, and then search through its isometric deformations".