Using the gauge theoretic approach for Lie applicable surfaces, we characterise certain subclasses of surfaces in terms of polynomial conserved quantities. These include isothermic and Guichard surfaces of conformal geometry and L-isothermic surfaces of Laguerre geometry. In this setting one can see that the well known transformations available for these surfaces are induced by the transformations of the underlying Lie applicable surfaces. We also consider linear Weingarten surfaces in this setting and develop a new Bäcklund-type transformation for these surfaces.
Burstall, F., Hertrich-Jeromin, U., Pember, M., & Rossman, W. (2019). Polynomial conserved quantities of Lie applicable surfaces. Manuscripta Mathematica, 158(3-4), 505-546. https://doi.org/10.1007/s00229-018-1033-0