  We explore a general framework how to treat coupled-channel systems in the
presence of overlapping left and right-hand cuts as well as anomalous
thresholds. Such systems are studied in terms of a generalized potential, where
we exploit the known analytic structure of t- and u-channel forces as the
exchange masses get smaller approaching their physical values. Given an
approximate generalized potential the coupled-channel reaction amplitudes are
defined in terms of non-linear systems of integral equations. For large
exchange masses, where there are no anomalous thresholds present, conventional
$N/D$ methods are applicable to derive numerical solutions to the latter. At a
formal level a generalization to the anomalous case is readily formulated by
use of suitable contour integrations with amplitudes to be evaluated at complex
energies. However, it is a considerable challenge to find numerical solutions
to anomalous systems set up on a set of complex contours. By a suitable
deformations of left-hand and right-hand cut lines we managed to establish a
framework of linear integral equations defined for real energies. Explicit
expressions are derived for the driving terms that hold for an arbitrary number
of channels. Our approach is illustrated in terms of a schematic 3-channel
systems. It is demonstrated that despite the presence of anomalous thresholds
the scattering amplitude can be represented in terms of 3 phase shifts and 3
in-elasticity parameters, as one would expect from the coupled-channel
unitarity condition.
