**Abstract: ** We study the asymptotic distribution of discrete
eigenvalues near the bottom of the essential spectrum for two
dimensional Pauli operator with magnetic field perturbed by electric
fields falling off at infinity. Unperturbed Pauli operator has the zero
energy as an isolated eigenvaue with infinite multiplicity if the magnetic
field is bounded from below by some positive constant. We first study
the asymptotic distribution of discrete eigenvalues around the
zero eigenvalue in this case. We consider also the case of the
magnetic field which converges to 0 at infinity.