Super regularity for Beltrami systems

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.