  By employing Aronsson's Absolute Minimizers of $L^\infty$ functionals, we
prove that Absolutely Minimizing Maps $u:\R^n \larrow \R^N$ solve a
"tangential" Aronsson PDE system. By following Sheffield-Smart \cite{SS}, we
derive $\De_\infty$ with respect to the dual operator norm and show that such
maps miss information along a hyperplane when compared to Tight Maps. We
recover the lost term which causes non-uniqueness and derive the complete
Aronsson system which has \emph{discontinuous coefficients}. In particular, the
Euclidean $\infty$-Laplacian is $\De_\infty u = Du \ot Du : D^2u\, +\,
|Du|^2[Du]^\bot \De u$ where $[Du]^\bot$ is the projection on the null space of
$Du^\top$. We exibit $C^\infty$ solutions having interfaces along which the
rank of their gradient is discontinuous and propose a modification with $C^0$
coefficients which admits \emph{varifold solutions}. Away from the interfaces,
Aronsson Maps satisfy a structural property of local splitting to 2 phases, an
horizontal and a vertical; horizontally they possess gradient flows similar to
the scalar case and vertically solve a linear system coupled by a scalar
Hamilton Jacobi PDE. We also construct singular $\infty$-Harmonic local $C^1$
diffeomorphisms and singular Aronsson Maps.
