Higher order parallel surfaces in three-dimensional homogeneous spaces

We give a full classification of higher order parallel surfaces in three-dimensional homogeneous spaces with four-dimensional isometry group, i.e. in the so-called Bianchi-Cartan-Vranceanu family. This gives a positive answer to a conjecture formulated in 2002. As a partial result, we prove that totally umbilical surfaces only exist if the space is a Riemannian product of a surface of constant Gaussian curvature and the real line, and we give a local parametrization of all totally umbilical surfaces.