  We employ a scalar model to exemplify the use of contour deformations when
solving Lorentz-invariant integral equations for scattering amplitudes. In
particular, we calculate the onshell 2 -> 2 scattering amplitude for the scalar
system. The integrals produce branch cuts in the complex plane of the integrand
which prohibit a naive Euclidean integration path. By employing contour
deformations, we can also access the kinematical regions associated with the
scattering amplitude in Minkowski space. We show that in principle a
homogeneous Bethe-Salpeter equation, together with analytic continuation
methods such as the Resonances-via-Pad\'e method, is sufficient to determine
the resonance pole locations on the second Riemann sheet. However, the scalar
model investigated here does not produce resonance poles above threshold but
instead virtual states on the real axis of the second sheet, which pose
difficulties for analytic continuation methods. To address this, we calculate
the scattering amplitude on the second sheet directly using the two-body
unitarity relation which follows from the scattering equation.
