  Given a finite collection of $C^1$ vector fields on a $C^2$ manifold which
span the tangent space at every point, we consider the question of when there
is locally a coordinate system in which these vector fields are real analytic.
We give necessary and sufficient, coordinate-free conditions for the existence
of such a coordinate system. Moreover, we present a quantitative study of these
coordinate charts. This is the third part in a three-part series of papers. The
first part, joint with Stovall, lay the groundwork for the coordinate system we
use in this paper and showed how such coordinate charts can be viewed as
scaling maps for sub-Riemannian geometry. The second part dealt with the
analogous questions with real analytic replaced by $C^\infty$ and Zygmund
spaces.
