The study of step 2 subRiemannian manifolds has the Heisenberg group as a prototype. This is a noncommutative Lie group with the base manifold R and endowed with a nonintegrable distribution spanned by two of the noncommutative left invariant vector fields. This structure enjoys also the property of being a contact structure or a CR-manifold. The study of the subRiemannian geodesics on the Heis...

‎In this paper‎, ‎we study translation invariant surfaces in the‎ ‎3-dimensional Heisenberg group $rm Nil_3$‎. ‎In particular‎, ‎we‎ ‎completely classify translation invariant surfaces in $rm Nil_3$‎ ‎whose position vector $x$ satisfies the equation $Delta x = Ax$‎, ‎where $Delta$ is the Laplacian operator of the surface and $A$‎ ‎is a $3 times 3$-real matrix‎.