We study the existence of solutions homoclinic to a saddle centre in a family of singularly perturbed fourth order differential equations, originating from a water-wave model. Due to a reversibility symmetry, the occurrence of such embedded solitons is a codimension-1 phenomenon. By varying a parameter a countable family of solitary waves is found. We examine the asymptotic frequency at which this phenomenon of persistence in the singular limit occurs, by performing a refined Stokes line analysis. In the limit where the parameter tends to infinity, each Stokes line splits into a pair, and the contributions of these two Stokes lines cancel each other for a countable set of parameter values. More generally, we derive the full leading order asymptotics for the Stokes constant, which governs the (exponentially small) amplitude of the (minimal) oscillations in the tails of nearly homoclinic solutions. True homoclinic trajectories are characterized by the Stokes constant vanishing. This formal asymptotic analysis is supplemented with numerical calculations. © 2010 IOP Publishing Ltd and London Mathematical Society.