  We study a class of static spherically symmetric vacuum solutions in modified
teleparallel gravity solving the field equations for a specific model Ansatz,
requiring the torsion scalar $T$ to be constant. We discuss the models falling
in this class. After some general considerations, we provide and investigate
local solutions in the form of black holes and traversable wormholes as well as
configurations that can match the anomalous rotation curves of galaxies.
