Singularities of the eigenvalue functions for first order symmetric symbols on rank two vector bundles over surfaces

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 $\sigma$.