There is a substantial literature on horizontally propagating waves in stratified flows. The vast majority of this work, particularly when it concerns nonlinear structures and solitary waves, focuses on “mode one”, that is, the "fast" waves of the system whereby all the pycnoclines (density jumps) are deflected with the same polarity. The simplest model for mode one waves is the two-layer flow of a lighter fluid above a heavier one bounded above and below by rigid boundaries. In that case the mode one wave is the only wave in the system. Mode two waves require (at least) one additional layer in order for the two interfaces to deflect with opposite polarity (mode two), or the same polarity (mode one). We shall consider the three-layer problem in this talk, considering KdV and MCC-like models, and the full Euler equations. We shall describe the problem and consider the question: do mode two solitary waves exist in the Euler equations?