  We construct linearized solutions to Vasiliev's four-dimensional higher spin
gravity on warped $AdS_3 \times_\xi S^1$ which is an $Sp(2)\times U(1)$
invariant non-rotating BTZ-like black hole with $\mathbb{R}^2\times T^2$
topology. The background can be obtained from $AdS_4$ by means of
identifications along a Killing boost $K$ in the region where $\xi^2\equiv
K^2\geqslant 0$, or, equivalently, by gluing together two
Ba\~nados--Gomberoff--Martinez eternal black holes along their past and future
space-like singularities (where $\xi$ vanishes) as to create a periodic
(non-Killing) time. The fluctuations are constructed from gauge functions and
initial data obtained by quantizing inverted harmonic oscillators providing an
oscillator realization of $K$ and of a commuting Killing boost $\widetilde K$.
The resulting solution space has two main branches in which $K$ star commutes
and anti-commutes, respectively, to Vasiliev's twisted-central closed two-form
$J$. Each branch decomposes further into two subsectors generated from ground
states with zero momentum on $S^1$. We examine the subsector in which $K$
anti-commutes to $J$ and the ground state is $U(1)_K\times U(1)_{\widetilde
K}$-invariant of which $U(1)_K$ is broken by momenta on $S^1$ and
$U(1)_{\widetilde K}$ by quasi-normal modes. We show that a set of
$U(1)_{\widetilde K}$-invariant modes (with $n$ units of $S^1$ momenta) are
singularity-free as master fields living on a total bundle space, although the
individual Fronsdal fields have membrane-like singularities at $\widetilde
K^2=1$. We interpret our findings as an example where Vasiliev's theory
completes singular classical Lorentzian geometries into smooth higher spin
geometries.
