  In this paper, we investigate mode-2 solitary waves in a three-layer
stratified flow model. Localised travelling wave solutions to both the fully
nonlinear problem (Euler equations), and the three-layer Miyata-Choi-Camassa
equations are found numerically and compared. Mode-2 solitary waves with speeds
slower than the linear mode-1 long-wave speed are typically generalised
solitary waves with infinite tails consisting of a resonant mode-one periodic
wave train. Herein we evidence the existence of mode-2 embedded solitary waves,
that is, we show that for specific values of the parameters, the amplitude of
the oscillations in the tail are zero. For sufficiently thick middle layers, we
also find branches of mode-2 solitary waves with speeds that extend beyond the
mode-1 linear waves and are no longer embedded. In addition, we show how large
amplitude embedded solitary waves are intimately linked to the conjugate states
of the problem.
