A class of surfaces related to a problem posed by \'{E}lie Cartan

We introduce a class of surfaces in euclidean space motivated by a problem posed by \'{E}lie Cartan. This class furnishes what seems to be the first examples of pairs of non-congruent surfaces in euclidean space such that, under a diffeomorphism $\Phi$, lines of curvatures are preserved and principal curvatures are switched. We show how to construct such surfaces using holomorphic data and discuss their relation with minimal surfaces. We also prove that if the diffeomorphism $\Phi$ preserves the conformal class of the third fundamental form, then all examples belong to the class of surfaces that we deal with in this work.