  For a rank two bundle $F$ over a surface $X$, we study the set of
singularities $M_\sigma$ of the eigenvalue functions of symmetric symbols
$\sigma$ associated to first order differential operators on $F$. We prove that
the existence of these singularities follows from topological considerations.
We define the degree of the set of singularities and show that it can be used
to count the number of directions at which special optical phenomena occur. For
the case when $F=TX$ we compute a formula for the degree in terms of the Euler
characteristics of $X$ and $N_\sigma$, where $N_\sigma\subset X$ is a manifold
with boundary associated to $\sigma$.
