  We prove a surprising higher regularity for solutions to the nonlinear
elliptic autonomous Beltrami equation in a planar domain $\Omega$, \[ f_\zbar =
{\cal A}(f_z) \hskip15pt a.e.\;\; z\in \Omega, \] when ${\cal A}$ is linear at
$\infty$. Namely $W^{1,1}_{loc}(\Omega)$ solutions are
$W^{2,2+\epsilon}_{loc}(\Omega)$. Here $\epsilon>0$ depends explicitly on the
ellipticity bounds of ${\cal A}$. The condition ``is linear at $\infty$'' is
necessary - the result is false for the equation $f_\zbar = k|f_z|$, for any
$0<k<1$, ($k=0$ is Weyl's lemma).
  We discuss the subsequent higher regularity implications for fully non-linear
Beltrami systems.
