  Geroch's solution-generating method is extended to the case of Einstein
spaces, which possess a Killing vector {{}and are thus asymptotically (locally)
(anti-)de Sitter}. This includes the reduction to a three-dimensional coset
space, the description of the dynamics in terms of a sigma-model and its
transformation properties under the $SL(2,\mathbb{R})$ group, and the
reconstruction of new four-dimensional Einstein spaces. The detailed analysis
of the space of solutions is performed using the Hamilton--Jacobi method in the
instance where the three-dimensional coset space is conformal to
$\mathbb{R}\times \mathcal{S}_2$. The cosmological constant appears in this
framework as a constant of motion and transforms under $SL(2,\mathbb{R})$.
