Position of the elliptic branches 1, 2, and 3 (see Table 1) of equilibria of Hamiltonian (25) in the resonance 1 : 1 : 2, for −2 ≤ δ ≤ 9. For each branch three curves appear, corresponding to for j = 1,2,3. The planetary masses are the same as in Table 1. A zoomed-in image close to the origin is shown in the inset. In this area, the analytical position of these equilibria, given by Eqs. (30) and (35), is plotted by a thin grey line. The agreement is good at low values of eccentricity, but quickly worsens further from the origin. In particular, the thin grey lines are straight since σ_j· does not depend on δ in Eq. (30). Branch 1 exists for all values of δ and has all colours from yellow to dark purple, while branch 3 only exists at δ > 5.997 and thus only has purple.