ON INVARIANT TORI IN SOME REVERSIBLE SYSTEMS

The paper proves Theorem 1.1 for the reversible system ẋ = ω0 + f(x,y), ẏ = g(x,y) with Diophantine ω0 and f=O(y), g=O(y^2), using a formal Birkhoff normal form (BNF) around Γ0 and a KAM counterterm (modifying-terms) scheme. It establishes: (i) positive-measure, density-one accumulation of invariant tori under 0-degeneracy; (ii) under j-degeneracy and the symmetry assumption (1.6), a (d+j)-dimensional analytic subvariety foliated by iso-frequency tori; and (iii) under (d−1)-degeneracy, a full neighborhood foliated by tori with frequencies proportional to ω0. These statements and their proofs are stated explicitly in the paper’s Theorem 1.1 and subsequent sections, including the BNF existence/uniqueness, the counterterm construction, and the transversality/density-one parameter estimates . The candidate solution follows the same overall architecture—formal BNF, a‑posteriori reversible KAM with modifying terms, Rüssmann-type parameter selection, and special treatment of j-degeneracies—but differs in details: it informally treats the BNF map f^F as analytic (where the paper emphasizes its formal nature) and appeals to an implicit-function argument in (ii) that the paper circumvents via the counterterm identity. These are presentation-level differences; the conclusions (i)–(iii) align with the paper’s results.