  In 1977 L\"uscher found a class of SO(4)-symmetric SU(2) Yang-Mills solutions
in Minkowski space, which have been rederived 40 years later by employing the
isometry $S^3\cong\mathrm{SU}(2)$ and conformally mapping SU(2)-equivariant
solutions of the Yang-Mills equations on (two copies of) de Sitter space
$\mathrm{dS}_4\cong\mathbb{R}{\times}S^3$. Here we present the noncompact
analog of this construction via $\mathrm{AdS}_3\cong\mathrm{SU}(1,1)$. On (two
copies of) anti-de Sitter space
$\mathrm{AdS}_4\cong\mathbb{R}{\times}\mathrm{AdS}_3$ we write down
SU(1,1)-equivariant Yang-Mills solutions and conformally map them to
$\mathbb{R}^{1,3}$. This yields a two-parameter family of exact SU(1,1)
Yang-Mills solutions on Minkowski space, whose field strengths are essentially
rational functions of Cartesian coordinates. Gluing the two AdS copies happens
on a $\mathrm{dS}_3$ hyperboloid in Minkowski space, and our Yang-Mills
configurations are singular on a two-dimensional hyperboloid
$\mathrm{dS}_3\cap\mathbb{R}^{1,2}$. This renders their action and the energy
infinite, although the field strengths fall off fast asymptotically except
along the lightcone. We also construct Abelian solutions, which share these
properties but are less symmetric and of zero action.
