Gerochs 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 $SL(2,mathbb{R})$.
