Real parts of the eigenvalues of the linearised system associated with F in the vicinity of its main branch of pseudo equilibria, for −7 ≤ δ ≤ 1 (left) and −6.2 ≤ δ ≤ −5 (right). The planetary masses are as in Table 1 and the resonance chain is 1:1:2. Only the four eigenvalues associated with the four degrees of freedom of the conservative system are represented. The other eigenvalues (like those associated with the rotation rates ω_j) are of no interest. The real parts behave erratically at the 1:1 secular resonance between ν and ν₃ (see Fig. 1) in such a way that all of them are negative for −5.64 ≤ δ ≤ −5.47, yielding a linear stability in this region. For δ ≤ −0.104, at least three out of four real parts are negative. In the region δ ≤ 1 we have , and the criterion (58) is very well respected. Similar figures with other planetary masses are available in Appendix F.