  General properties of eigenvalues of $A+\tau uv^*$ as functions of
$\tau\in\Comp$ or $\tau\in\Real$ or $\tau=\e^{\ii\theta}$ on the unit circle
are considered. In particular, the problem of existence of global analytic
formulas for eigenvalues is addressed. Furthermore, the limits of eigenvalues
with $\tau\to\infty$ are discussed in detail. The following classes of matrices
are considered: complex (without additional structure), real (without
additional structure), complex $H$-selfadjoint and real $J$-Hamiltonian.
