Solitons with fermion number ½

We study the structure of soliton-monopole systems when Fermi fields are present. We show that the existence of a nondegenerate, isolated, zero-energy, $c$-number solution of the Dirac equation implies that the soliton is a degenerate doublet with Fermi number ± ½. We find such solutions in the theory of Yang-Mills monopoles and dyons.