  We consider the resolvent of a system of first order differential operators
with a regular singularity, admitting a family of self-adjoint extensions. We
find that the asymptotic expansion for the resolvent in the general case
presents powers of $\lambda$ which depend on the singularity, and can take even
irrational values. The consequences for the pole structure of the corresponding
$\zeta$ and $\eta$-functions are also discussed.
