  We consider a helicoidal group $G$ in $\mathbb{R}^{n+1}$ and unbounded
$G$-invariant $C^{2,\alpha}$-domains $\Omega\subset\mathbb{R}^{n+1}$ whose
helicoidal projections are exterior domains in $\mathbb{R}^{n}$, $n\geq2$. We
show that for all $s\in\mathbb{R}$, there exists a $G$-invariant solution
$u_{s}\in C^{2,\alpha}\left( \overline{\Omega}\right) $ of the Dirichlet
problem for the minimal surface equation with zero boundary data which
satisfies $\sup_{\partial\Omega}\left\vert \operatorname{grad}u_{s}\right\vert
=\left\vert s\right\vert $. Additionally, we provide further information on the
behavior of these solutions at infinity.
