  We investigate solutions $(\mathcal{M}, g)$ to Einstein's vacuum field
equations with positive cosmological constant $\Lambda$ which admit a smooth
past null infinity $\mathcal{J}^-$ \`a la Penrose and a Killing vector field
whose associated Mars-Simon tensor (MST) vanishes. The main purpose of this
work is to provide a characterization of these spacetimes in terms of their
Cauchy data on $\mathcal{J}^-$.
  Along the way, we also study spacetimes for which the MST does not vanish. In
that case there is an ambiguity in its definition which is captured by a scalar
function $Q$. We analyze properties of the MST for different choices of $Q$. In
doing so, we are led to a definition of "asymptotically Kerr-de Sitter-like
spacetimes", which we also characterize in terms of their asymptotic data on
$\mathcal{J}^-$.
