  We analyze a certain class of integral equations related to Marchenko
equations and Gel'fand-Levitan equations associated with various systems of
ordinary differential operators. When the integral operator is perturbed by a
finite-rank perturbation, we explicitly evaluate the change in the solution. We
show how this result provides a unified approach to Darboux transformations
associated with various systems of ordinary differential operators. We
illustrate our theory by deriving the Darboux transformation for the
Zakharov-Shabat system and show how the potential and wave function change when
a discrete eigenvalue is added to the spectrum.
