  Let S be a smooth real surface in C^2 and let p\in S be a point at which the
tangent plane is a complex line. How does one determine whether or not S is
locally polynomially convex at such a p --- i.e. at a CR singularity ? Even
when the order of contact of T_p(S) with S at p equals 2, no clean
characterisation exists; difficulties are posed by parabolic points. Hence, we
study non-parabolic CR singularities. We show that the presence or absence of
Bishop discs around certain non-parabolic CR singularities is completely
determined by a Maslov-type index. This result subsumes all known facts about
Bishop discs around order-two, non-parabolic CR singularities. Sufficient
conditions for Bishop discs have earlier been investigated at CR singularities
having high order of contact with T_p(S). These results relied upon a
subharmonicity condition, which fails in many simple cases. Hence, we look
beyond potential theory and refine certain ideas going back to Bishop.
