Classification of Kerr-de Sitter-like spacetimes with conformally flat Scri

We provide a classification of $\mathrm{\Lambda }>0$-vacuum spacetimes which admit a Killing vector field with respect to which the associated “Mars-Simon tensor” (MST) vanishes and having a conformally flat ${\mathcal{J}}^{-}$ (or ${\mathcal{J}}^{+}$). To that end, we also give a complete classification of conformal Killing vector fields on the $3$-sphere ${\mathbb{S}}^3$ up to M"obius transformations shedding light on the two fundamental constants that characterize the family of Kerr-de Sitter-like spacetimes, which turn out to be well-defined geometrical invariants. The topology of $\mathcal{J}$ is determined in every case, and a characterization result at $\mathcal{J}$ of the Kerr-de Sitter family presented.