  Motivated by the existence of candidates for exotic hadrons whose masses are
close to both of two-body and three-body hadronic thresholds lying close to
each other, we study degenerate two-body and three-body coupled-channel
systems. We first formulate the scattering problem of non-degenerate two-body
and three-body coupled-channels as an effective three-body problem, i.e.\
effective Alt-Grassberger-Sandhas (AGS) equations. We next investigate the
behavior of $S$-matrix poles near the threshold when two-body and three-body
thresholds are degenerate. We solve the eigenvalue equations of the kernel of
AGS equations instead of AGS equations themselves to obtain the $S$-matrix pole
energy. We then face a problem of unphysical singularity: though the physical
transition amplitudes have physical singularities only, the kernel of AGS
equations have unphysical singularities. We show, however, that these
unphysical singularities can be removed by appropriate reorganization of the
scattering equations and mass renormalization. The behavior of $S$-matrix poles
near the degenerate threshold is found to be universal in the sense that the
complex pole energy, $E$, is determined by a real parameter, $c$, as $c - E
\log{\left( - E \right)} = 0$, or equivalently, ${\rm Im} E = - \pi {\rm Re} E
/ \log{\mid {\rm Re} E \mid}$. This behavior is different from that of either
two-body or three-body system and is characteristic in the degenerate two-body
and three-body coupled-channel system. We expect that this new class of
universal behavior might play a key role in understanding exotic hadrons.
