In this paper, we study the travelling gravity waves in a system of two layers of perfect fluids, the bottom one being infinitely deep, the upper one having a finite thickness h. We assume that the flow is potential, and the dimensionless parameters are the ratio between densities ρ = ρ 2 /ρ 1 and λ = gh/c^2. We study special values of the parameters such that λ(1 − ρ) is near 1 − , where a bifurcation of a new type occurs. We formulate the problem as a spatial reversible dynamical system, where U = 0 corresponds to a uniform state (velocity c in a moving reference frame), and we consider the linearized operator around 0. We show that its spectrum contains the entire real axis (essential spectrum), with in addition a double eigenvalue in 0, a pair of simple imaginary eigenvalues ±iλ at a distance O(1) from 0, and for λ(1 − ρ) above 1, another pair of simple imaginary eigenvalues tending towards 0 as λ(1 − ρ) → 1 +. When λ(1 − ρ) ≤ 1 this pair disappears into the essential spectrum. The rest of the spectrum lies at a distance at least O(1) from the imaginary axis. We show in this paper that for λ(1 − ρ) close to 1 − , there is a family of periodic solutions like in the Lyapunov-Devaney theorem (despite the resonance due to the point 0 in the spectrum). Moreover, showing that the full system can be seen as a perturbation of the Benjamin-Ono equation, coupled with a nonlinear oscillation, we also prove the existence of a family of homoclinic connections to these periodic orbits, provided that these ones are not too small.