  In this paper we study properties that the vacuum must possess in the minimal
extension to the teleparallel equivalent of general relativity (TEGR) where the
action is supplemented with a quadratic torsion term. No assumption is made
about the weakness of the quadratic term although in the weak-field regime the
validity of our previously derived perturbative solution is confirmed.
Regarding the exact nature of the vacuum, it is found that if the center of
symmetry is to be regular, the mathematical conditions on the tetrad at the
isotropy point mimic those of general relativity. With respect to horizons it
is found that, under very mild assumptions, a smooth horizon cannot exist
unless the quadratic torsion coupling, $\alpha$, vanishes, which is the TEGR
limit (with the Schwarzschild tetrad as its solution). This analysis is then
supplemented with computational work utilizing asymptotically Schwarzschild
boundary data. It is verified that in no case studied does a smooth horizon
form. For $\alpha > 0$ naked singularities occur which break down the equations
of motion before a horizon can form. For $\alpha < 0$ there is a limited range
of $\alpha$ where a vacuum horizon might exist but, if present, the horizon is
singular. Therefore physically acceptable black hole horizons are problematic
in the studied theory at least within the realm of vacuum static spherical
symmetry. These results also imply that static spherical matter distributions
generally must have extra restrictions on their spatial extent and
stress-energy bounds so as to render the vacuum solution invalid in the
singular region and make the solutions finite.
