  We find a large family of solutions to the Dirac equation on a manifold of
$G_2$ holonomy asymptotic to a cone over $S^3 \times S^3$, including all radial
solutions. The behaviour of these solutions is studied as the manifold
developes a conical singularity. None of the solutions found are both localised
and square integrable at the origin. This result is consistent with the absence
of chiral fermions in M-theory on the conifold over $S^3\times S^3$. The
approach here is complementary to previous analyses using dualities and anomaly
cancellation.
