Secular dynamics for curved two-body problems

The paper proves two statements for the reduced curved two-body problem in the small-separation regime: a positive-measure Cantor family of KAM tori with Diophantine frequencies (Theorem 1) and the existence of long-period periodic orbits with prescribed slow precession count (Theorem 2). It builds a high-order secular normal form (Prop. 4.1) and then applies a quantitative KAM theorem à la Féjoz to obtain quasi-periodic tori (Theorem 1), and uses an implicit-function argument on a long-time return map based on the order-ε expansions of the averaged term 〈Per〉 (eq. (10)) to get the periodic orbits (Theorem 2) . The candidate solution follows the same high-order averaging/KAM architecture and checks Kolmogorov nondegeneracy explicitly, but proves periodic orbits via a twist-map/Poincaré–Birkhoff route rather than the paper’s implicit-function route. Aside from a minor sign slip in the L–L Hessian entry and a slightly different order bookkeeping for the remainder, the model’s arguments align with the paper’s framework and yield the same conclusions.