  We investigate the relativistic scattering of three identical scalar bosons
interacting via pair-wise interactions. Extending techniques from the
non-relativistic three-body scattering theory, we provide a detailed and
general prescription for solving and analytically continuing integral equations
describing the three-body reactions. We use these techniques to study a system
with zero angular momenta described by a single scattering length leading to a
bound state in a two-body sub-channel. We obtain bound-state--particle and
three-particle amplitudes in the previously unexplored kinematical regime; in
particular, for real energies below elastic thresholds and complex energies in
the physical and unphysical Riemann sheets. We extract positions of
three-particle bound-states that agree with previous finite-volume studies,
providing further evidence for the consistency of the relativistic
finite-volume three-body quantization conditions. We also determine previously
unobserved virtual bound states in this theory. Finally, we find numerical
evidence of the breakdown of the two-body finite-volume formalism in the
vicinity of the left-hand cuts and argue for the generalization of the existing
formalism.
