We give examples of systems of Partial Differential Equations that admit non-trivial, Lipschitz and one-homogeneous solutions in the form u(R, θ) = Rg(θ), where (R, θ) are plane polar coordinates and g :The systems are singular in the sense that they arise as the Euler-Lagrange equations of the functionals|x| as |x| → 0 and W satisfies an ellipticity condition. Such solutions cannot exist when |x|D F W (x, F ) → 0 as |x| → 0, so the condition is optimal. The associated analysis exploits the well-known Fefferman-Stein duality [7]. We also discuss conditions for the uniqueness of these one-homogeneous solutions and demonstrate that they are minimizers of certain variational functionals.