  We give examples of systems of Partial Differential Equations that admit
non-trivial, Lipschitz and one-homogeneous solutions in the form $u(R,\theta) =
Rg(\theta)$, where $(R,\theta)$ are plane polar coordinates and $g:
\mathbb{R}^{2} \to \mathbb{R}^{m}$, $m \geq 2$. The systems are singular in the
sense that they arise as the Euler-Lagrange equations of the functionals $I(u)
= \int_{B}W(x,\nabla u(x))\,dx$, where $D_{F}W(x,F)$ behaves like
$\frac{1}{|x|}$ as $|x| \to 0$ and $W$ satisfies an ellipticity condition. Such
solutions cannot exist when $|x|D_{F}W(x,F) \to 0$ as $|x| \to 0$, so the
condition is optimal. The associated analysis exploits the well-known
Fefferman-Stein duality. We also discuss conditions for the uniqueness of these
one-homogeneous solutions and demonstrate that they are minimizers of certain
variational functionals.
