INTEGRABLE SYSTEMS ON SINGULAR SYMPLECTIC MANIFOLDS: FROM LOCAL TO GLOBAL

The paper establishes a folded action–angle theorem (Theorem 14) and derives the normal form ω = τ dθ1 ∧ dτ + ∑_{i=2}^n dθi ∧ dσ_i around a regular Liouville torus inside the folding hypersurface Z, via period-uniformization, an equivariant relative Poincaré lemma, and a folded Darboux–Carathéodory theorem; see the statement (Theorem 14) and the concluding normal form in the proof, as well as the supporting lemmas (Proposition 10, Theorem 11) . The candidate solution produces the same normal form by an Arnold–Liouville style argument: constructing a local T^n-action, using closed/basic 1-forms ι_{Y_i}ω to define base functions, identifying g1 = ½τ^2, and canceling a basic 2-form remainder by a base-dependent angle shift. The approach is correct and matches the paper's result, though it sketches (rather than proves) the period uniformization and the isotropy/basicness needed to ensure ι_{Y_j}ι_{Y_i}ω = 0 across Z (handled carefully in the paper via the uniformization construction and averaging) .