When do Lyapunov Subcenter Manifolds become Eigenmanifolds?

The paper proves (i) existence/uniqueness of a 2D Lyapunov subcenter manifold (LSM) near an elliptic equilibrium under standard analytic and nonresonance hypotheses; (ii) that this LSM is a weak Eigenmanifold and contains a local Eigenmanifold; (iii) in dim(Q)=2 the whole LSM is an Eigenmanifold; and (iv) with an additional spatial involution φ satisfying M=φ∗M and V=V∘φ, one obtains (weak) Rosenberg manifolds, global under a uniqueness-of-fixed-point condition. These are Theorems 23, 25, 26, and 28, supported by Theorem 21 and classical LSM theory (Theorem 32) . The candidate solution reproduces the same results via a standard parameterization/cohomological-equation approach, together with symmetry/reversibility arguments. Two minor issues: (a) in part (3) their use of invariance of domain to claim a global embedding of the annulus lacks a proof of injectivity (the paper supplies a different, correct argument excluding self-intersections in 2D) ; (b) they tacitly use a “generator” property for general LSMs that in the paper is established for Eigenmanifolds (not arbitrary LSMs) . These are repairable without changing the conclusions. Overall, both are correct; the proofs differ in method and the model’s proof needs minor polishing.