  We analyze the supersymmetry transformations of gauged $SO(4)$ supergravity
coupled to extra hypermultiplets in three dimensions, and find large families
of smooth BPS solutions that preserve four supersymmetries. These BPS solutions
are part of the consistent truncation of some families of six-dimensional
superstrata. From the three-dimensional perspective, these solutions give rise
to "smoothly-capped BTZ" geometries. We show how the twisting of the spin
connection, the holomorphy of the fields, and the Chern-Simons connections all
play an essential role in the existence of these supersymmetric solutions. This
paper also closes the circle on the consistent truncation of superstrata,
showing precisely how every feature of the superstratum enters into the
three-dimensional BPS structure.
