  For $\mathrm{H} \in C^2(\mathbb{R}^{N \times n})$ and $u : \Omega \subseteq
\mathbb{R}^n \to \mathbb{R}^N$, consider the system \[
\label{1}\mathrm{A}\_\infty u\, :=\,\Big(\mathrm{H}\_P \otimes \mathrm{H}\_P +
\mathrm{H}[\mathrm{H}\_P]^\bot \mathrm{H}\_{PP}\Big)(\mathrm{D} u):
\mathrm{D}^2 u\, =\,0. \tag{1}\]We construct $\mathcal{D}$-solutions to the
Dirichlet problem for (1), an apt notion of generalised solutions recently
proposed for fully nonlinear systems. Our $\mathcal{D}$-solutions are
$W^{1,\infty}$-submersions and are obtained without any convexity hypotheses
for $\mathrm{H}$, through a result of independent interest involving existence
of strong solutions to the singular value problem for general dimensions $n\neq
N$.

Analysis of PDEs