  The Marchenko method is developed in the inverse scattering problem for a
linear system of first-order differential equations containing potentials
proportional to the spectral parameter. The corresponding Marchenko system of
integral equations is derived in such a way that the method can be applied to
some other linear systems for which a Marchenko method is not yet available. It
is shown how the potentials and the scattering solutions to the linear system
are constructed from the solution to the Marchenko system. The bound-state
information for the linear system with any number of bound states and any
multiplicities is described in terms of a pair of constant matrix triplets.
When the potentials in the linear system are reflectionless, some explicit
solution formulas are presented in closed form for the potentials and for the
scattering solutions to the linear system. The theory is illustrated with some
explicit examples.
