  We analyze zero energy solutions of the Dirac equation in the background of a
string-like configuration in an extension of the standard model which
accommodates the most general fermionic mass matrix for neutrinos. If either
the left- or the right-handed Majorana mass vanishes, neutrino and electron
zero modes are found to exist. If the harmonic mean of the neutrino Majorana
masses is large compared to the Dirac mass, we can prove that normalizable
neutrino zero modes cease to exist. This leads to an odd number of fermionic
zero modes -- namely, only the electron zero mode -- which we argue implies
that the bosonic background is in a topologically distinct sector from the
vacuum. This is confirmed by noting that the bosonic sector of the model
involves the breaking of a global U(1) symmetry and hence possesses topological
global U(1) strings.
