  In this paper we study massless fermions coupled to spherically symmetric
$SU(N)$ monopoles without Yukawa couplings between the Higgs and fermion
fields. The corresponding Dirac operator is not Fredholm and the associated
eigenfunctions are not $L^2$-normalizable. Here we derive a formula for the
dimension of the plane-wave normalizable kernel of such a Dirac operator for
fermions of any representation of $SU(N)$ in the presence of any spherically
symmetric monopole background. Notably, our results also apply to fermions
coupled to monopoles that preserve non-abelian gauge symmetry.
