  We show that the group of isometries (i.e., distance-preserving
homeomorphisms) of an equiregular subRiemannian manifold is a
finite-dimensional Lie group of smooth transformations. The proof is based on a
new PDE argument, in the spirit of harmonic coordinates, establishing that in
an arbitrary subRiemannian manifold there exists an open dense subset where all
isometries are smooth.
