Liouville theorem, conformally invariant cones and umbilical surfaces   for Grushin-type metrics
  We prove a classification theorem for conformal maps with respect to the control distance generated by a system of diagonal vector fields.   It turns out that all such maps can be obtained as compositions of suitable dilations, inversions and isometries. We also classify all umbilical surfaces of the underlying metric. 
Daniele Morbidelli