Covariant classification of conformal Killing vectors of locally conformally flat $n$-manifolds with an application to Kerr-de Sitter

We obtain a coordinate independent algorithm to determine the class of conformal Killing vectors of a locally conformally flat $n$-metric $\gamma$ of signature $(r,s)$ modulo conformal transformations of $\gamma$. This is done in terms of endomorphisms in the pseudo-orthogonal Lie algebra $\mathfrak{o}(r+1,s+1)$ up to conjugation of the its group $O(r+1,s+1)$. The explicit classification is worked out in full for the Riemannian $\gamma$ case ($r=0,s=n$). As an application of this result, we prove that the set of five dimensional, $(\mathrm{\Lambda }>0)$-vacuum, algebraically special metrics with non-degenerate optical matrix, previously studied by Bernardi de Freitas, Godazgar and Reall, is in one-to-one correspondence with the metrics in the Kerr-de Sitter-like class. This class exists in all dimensions and its defining properties involve only properties at $\mathcal{I}$. The equivalence between two seemingly unrelated classes of metrics points towards interesting connections between the algebraically special type of the bulk spacetime and the conformal geometry at null infinity