Hidden toric symmetry and structural stability of singularities in integrable systems

The paper’s Theorem 3.10 establishes a canonical linear model for a Hamiltonian torus action near a singular orbit (including the finite twisting group Γ, Williamson block structure, integer weight matrix with zero-pattern constraints, and the dimension inequalities κe ≤ ke+kf, κh ≤ kh+kf, r+κe+κh ≤ r+ke+kh+2kf ≤ n, with nondegeneracy when equality holds), and proves persistence under analytic perturbations; see the statement and proof outline in the general case and the fixed-point lemmas (Theorem 3.10 and its proof, plus (36)–(39)) . The candidate solution independently reconstructs this result via an MGS slice/symplectic-slice approach, simultaneous Williamson normal form, identification of the integer weight lattice, a Γ-action satisfying Property (L), and an equivariant Moser argument on the complexification for persistence. The approach differs in presentation but matches the paper’s conclusions and invariants, including Property (L)(iv) for inert factors and the same dimension bounds and nondegeneracy criterion. Minor gaps (e.g., a quick justification for the involutive behavior on inert factors) are covered in the paper’s detailed linear-algebraic lemmas. Overall, both are correct and compatible, with distinct proof organization .