The contact mappings of a flat $(2,3,5)$-distribution

Let $\Omega$ and $\Omega'$ be open subsets of a flat $(2,3,5)$-distribution. We show that a $C^1$-smooth contact mapping $f : \Omega \to \Omega'$ is a $C^\infty$-smooth contact mapping. Ultimately, this is a consequence of the rigidity of the associated stratified Lie group (the Tanaka prolongation of the Lie algebra is of finite-type). The conclusion is reached through a careful study of some differential identities satisfied by components of the Pansu-derivative of a $C^1$-smooth contact mapping.