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We present a gluing construction which adds, via a localized deformation, \emph{exactly Delaunay ends} to \emph{generic} metrics with constant positive scalar curvature. This provides time-symmetric initial data sets for the vacuum Einstein equations with positive cosmological constant with \emph{exactly} Kottler--Schwarzschild--de Sitter ends, extending the results in~\cite{ChPollack}.

Singular Yamabe metrics and initial data with \emph{exactly} Kottler–Schwarzschild–de Sitter ends II. Generic metrics