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

Let $\mathrm{\Omega }$ and ${\mathrm{\Omega }}^{\mathrm{\prime }}$ be open subsets of a flat $(2,3,5)$-distribution. We show that a $C^1$-smooth contact mapping $f:\mathrm{\Omega }\to {\mathrm{\Omega }}^{\mathrm{\prime }}$ is a $C^{\mathrm{\infty }}$-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.