  We consider a two-dimensional electron with an anomalous magnetic moment,
g>2, interacting with a nonzero magnetic field B perpendicular to the plane
which gives rise to a flux F. Recent results about the discrete spectrum of the
Pauli operator are extended to fields with the O(r^{-2-\delta}) decay at
infinity: we show that if |F| exceeds an integer N, there is at least N+1 bound
states. Furthermore, we prove that weakly coupled bound states exist under mild
regularity assumptions also in the zero flux case.
