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To date, studies of $\textit{Laplace Surface}$ dynamics have concernedthemselves with test particle orbits of fixed shape and orientation in thecombined field of an oblate central body (to which the particle is bound) and adistant, inclined, companion which is captured to quadrupolar order. Whileamply sufficient for satellites around planets on near-circular orbits, thequadrupolar approximation fails to capture essential dynamical features inducedby a wide binary companion (be it a star, a planet or a black hole) on a fairlyeccentric orbit. With similar such astronomical settings in mind, we extend theclassical Laplace framework to higher multipoles, and map out the backbone ofstationary orbits, now complexified by the broken axial symmetry. Eccentric andinclined Laplace equilibria, which had been presaged in systems of large enoughmutual inclination, are here delineated over a broad range of mutually inclinedperturbations. We recover them for test particles in the field of a hot Jupiterand a wide eccentric stellar binary, highlighting their relevance for thearchitecture of multi-planet systems in binaries. We then extend and deploy ourmachinery closer to home, as we consider the secular dynamics ofTrans-Neptunian Objects (TNOs) in the presence of a putative ninth planet. Weshow how generalized Laplace equilibria seed islands for Trans-Neptunianobjects to be sheltered around, islands within chaotic seas which we capturevia Poincar$\'{e}$ sections, while highlighting a beautiful interplay betweenLaplace and Kozai-Lidov secular dynamical structures. An eminently classicaltale revived for the exo-planetary 21st century!

Laplace surface dynamics, revisited: satellites, exo-planets and debris with distant, eccentric companions