  Starting from the Hawking--Page solutions, we consider the corresponding
Lorentzian cone metrics. These represent cone interior scale-invariant vacuum
solutions, defined in the chronological past of the scaling origin. We extend
the Lorentzian Hawking--Page solutions to the cone exterior region in the class
of $(4+1)$-dimensional scale-invariant vacuum solutions with an $SO(3)\times
U(1)$ isometry, using the Kaluza--Klein reduction and the methods of
Christodoulou. We prove that each Lorentzian Hawking--Page solution has
extensions with a null curvature singularity, extensions with a spacelike
curvature singularity, and extensions with a null Cauchy horizon of Taub--NUT
type. These are all the possible extensions within our symmetry class. The
extensions to spacetimes with a null curvature singularity can be used to
construct $(4+1)$-dimensional asymptotically flat vacuum spacetimes with
locally naked singularities, where the null curvature singularity is not
preceded by trapped surfaces. We prove the instability of such locally naked
singularities using the blue-shift effect of Christodoulou.
