  We consider a Bose-Einstein condensate, which is confined in a very tight
toroidal/annular trap, in the presence of a potential, which breaks the axial
symmetry of the Hamiltonian. We investigate the stationary states of the
condensate, when its density distribution co-rotates with the symmetry-breaking
potential. As the strength of the potential increases, we have a gradual
transition from vortex excitation to solid-body-like motion. Of particular
importance are states where the system is static and yet it has a nonzero
current/circulation, which is a realization of persistent
currents/reflectionless potentials.
