  In this work we define the Ribaucour-type surfaces (in short, RT-surfaces).
These surfaces satisfy a relationship similar to the Ribaucour surfaces that
are related to the \'Elie Cartan problem. This class furnishes what seems to be
the first examples of pairs of noncongruent surfaces in Euclidean space such
that, under a diffeomorphism, lines of curvatures are preserved and principal
curvatures are switched. We show that every compact and connected RT-surface is
a sphere with center at the origin. We obtain present a Weierstrass type
representation for RT-surfaces with prescribed Gauss map which depends on two
holomorphic functions. We give explicit examples of RT-surfaces. Also, we use
this representation to classify the RT-surfaces of rotation.
