  Asymptotically flat black holes in $2+1$ dimensions are a rarity. We study
the recently found black flower solutions (asymptotically flat black holes with
deformed horizons), static black holes, rotating black holes and the dynamical
black flowers (black holes with radiative gravitons ) of the purely quadratic
version of new massive gravity. We show how they appear in this theory and we
also show that they are also solutions to the infinite order extended version
of the new massive gravity, that is the Born-Infeld extension of new massive
gravity with an amputated Einsteinian piece. The same metrics also solve the
topologically extended versions of these theories, with modified conserved
charges and the thermodynamical quantities, such as the Wald entropy. Besides
these we find new conformally flat radiating type solutions to these extended
gravity models. We also show that these metrics do not arise in Einstein's
gravity coupled to physical perfect fluids.
