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Two semi-analytical one-degree-of-freedom secular models are presented forthe motion of small bodies beyond Neptune. A special attention is given totrajectories entirely exterior to the planetary orbits. The first one is thewell-known non-resonant model of Kozai (1962) adapted to the transneptunianregion. Contrary to previous papers, the dynamics is fully characterized withrespect to the fixed parameters. A maximum perihelion excursion possible of16.4 AU is determined. The second model handles the occurrence of a mean-motionresonance with one of the planets. In that case, the one-degree-of-freedomintegrable approximation is obtained by postulating the adiabatic invariance,and is much more general and accurate than previous secular models found in theliterature. It brings out in a plain way the possibility of perihelionoscillations with a very high amplitude. Such a model could thus be used infuture studies to deeper explore that kind of motion. For complex resonantorbits (especially of type 1:k), a segmented secular description is necessarysince the trajectories are only "integrable by parts". The two models areapplied to the Solar System but the notations are kept general so that it couldbe used for any quasi-circular and coplanar planetary system.

Long term dynamics beyond Neptune: secular models to study the regular motions