  In this paper we consider a set $E\subset\Omega$ with prescribed mean
curvature $f\in C(\Omega)$ and Euclidean Lipschitz boundary $\partial E=\Sigma$
inside a three-dimensional contact sub-Riemannian manifold $M$. We prove that
if $\Sigma$ is locally a regular intrinsic graph, the characteristic curves are
of class $C^2$. The result is shape and improves the ones contained in
\cite{MR2583494} and \cite{GalRit15}.
