Regularity of sets with constant horizontal normal in the Engel group

In the Engel group with its Carnot group structure, we study subsets of locally finite subRiemannian perimeter and possessing constant subRiemannian normal.We prove the rectifiability of such sets: more precisely we show that, in some specific coordinates, they are upper-graphs of entire Lipschitz functions (with respect to the Euclidean distance). However we find that, when they are written as intrinsic horizontal upper-graphs with respect to the direction of the normal, then the function defining the set might even fail to be continuous. Nevertheless, we can prove that one can always find other horizontal directions for which the set is the intrinsic horizontal upper-graph of a function that is Lipschitz-continuous with respect to the intrinsic subRiemannian cones (and in particular locally Hölder-continuous for the Euclidean distance). We further discuss a partial differential equation characterization of the class of all sets with constant horizontal normal. Finally, we show that our rectifiability argument extends to the case of filiform groups of the first kind.