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The high-inclination population in the 4:7 mean motion resonance (MMR) withNeptune has also substantial eccentricities ($e\gtrsim0.1$), with more inclinedobjects tending to occupy more eccentric orbits. For this high-order resonance,there are two different resonant modes. The principal one is theeccentricity-type mode, and we find that libration is permissible for orbitswith $e\ge e_c^0$, where the critical eccentricity $e_c^0$ increases as afunction of increasing inclination $i$. Correspondingly, we introduce alimiting curve $e_c^0(i)$, which puts constraints on the $(e, i)$ distributionof possible 4:7 resonators. We then perform numerical simulations on thesweep-up capture and long-term stability of the 4:7 MMR, and the results showthat the simulated resonators are well-constrained by this theoretical limitingcurve. The other 4:7 resonant mode is the mixed-$(e, i)$-type, and we show thatstable resonators should exist at $i\gtrsim20^{\circ}$. We predict that theintrinsic number of these mixed-$(e, i)$-type resonators may provide a new clueinto the Solar system's evolution, but, so far, only one real object has beenobserved resonating in this mode.

A study of the high-inclination population in the Kuiper belt -- III. The 4:7 mean motion resonance