Level curves of the Hamiltonian (29). The plotted energy levels correspond to the Hamiltonian characterized by the mass-ratio and angular momentum equivalent to that of the 2:1 resonant HD 82943 system (see Table 1 as well as Fig. 4). The four dynamical portraits depicted in panels labeled A) B) C) D) exhibit different proximities to exact resonance. Specifically, the associated values of Φ₂ are A) Φ₂ = 1.6 × 10^-4; B) Φ₂ = 4.8 × 10^-4; C) Φ₂ = 9.6 × 10^-4; D) Φ₂ = 1.12 × 10^-3. Note that the dynamical portraits shown in panels A), B) and C) feature the presence of separatrcies, shown as thick gray curves. The separatrix disappears in panel D), although libration of the critical angle is still possible, as depicted by the thick orange circle. Note the factor of two reduction in the axes ranges between panels A) and B) and C) and D).