  We provide a classification of $\Lambda>0$-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.
