Invariant Tori for Multi-Dimensional Integrable Hamiltonians Coupled to a Single Thermostat

The paper proves a normal form for the Nosé-thermostated, integrable Hamiltonian near the thermostatic equilibrium set and applies a properly-degenerate analytic KAM theorem under R-non-degeneracy of the re-scaled frequency map to obtain a positive-measure set of invariant tori for large mass Q. The candidate solution reproduces the same two pillars: (i) an explicit chain of exact symplectic transformations yielding H∘ϕ = Ḡ(Î) + (ε/2)α(Î)(v^2+V^2) + ε^{3/2}Rε with Rε analytic in ε>0 and continuous at ε=0, and (ii) a KAM persistence argument under Rüssmann non-degeneracy, providing positive-measure Lagrangian tori near the thermostatic equilibrium manifold. The steps, scalings, and the derived frequency map Ω = (dḠ, α) match the paper’s expressions (up to notational choices). Minor differences (e.g., the local coordinates used to flatten the manifold and the phrasing about the ε-threshold in T) do not affect correctness.