  In this paper we relax the current regularity theory for the eikonal equation
by using the recent theory of { set-valued} iterated Lie brackets. We give
sufficient conditions for small time local attainability of general, symmetric,
nonlinear systems, which have as a consequence the Hoelder regularity of the
minimum time function in optimal control. We then apply such result to prove
H\"older continuity of solutions of the Dirichlet boundary value problem for
the eikonal equation with low regularity of the coefficients. We also prove
that the sufficient conditions for the H\"older regularity are essentially
necessary, at least for smooth vector fields and target.
