That is an American bb shell.

The adaptor is eccentric bb.

Are you running single or multi speed?

If single speed, you can use
1. Bmx euro bb and crank with eccentric bb. Euro is simply bmx cranks and bb mounted in a standard English threaded bb shell.
2. Bmx American bb (simpler).

3. If multi speed, you can use any standard mtb (for english threaded bb shell eg square taper, Hollowtech 2) bb with eccentric bb.