NONINTEGRABILITY OF DYNAMICAL SYSTEMS NEAR DEGENERATE EQUILIBRIA

The paper proves real-analytic nonintegrability (in the Bogoyavlenskij sense) near degenerate equilibria in 3D (Hopf–zero) and 4D (double Hopf with incommensurate frequencies) by reducing the problem to planar truncated systems (2.11)–(2.12) and showing these have only non-analytic first integrals and no commuting vector fields, hence contradicting integrability via Theorem 2.4 and Zung’s normal-form result (Theorems 1.2–1.3; reduction and planar analysis in Sections 2–4) . The candidate solution instead gives a homological normal-form obstruction: any analytic first integral’s lowest-degree term must satisfy L_ℓP=0 and, because [ℓ,N]=0, also L_NP=0; a monomial-level analysis on S^1 or T^2 invariants then rules out nontrivial polynomial solutions under the same sign/irrationality hypotheses, and Frobenius precludes extra commuting fields (Definition 1.1 aligns with this logic) . The two approaches are compatible and reach the same conclusions. Minor caveat: in the 4D case, the candidate’s “triangular” argument implicitly uses a minimal-exponent induction to force the two linear constraints on each surviving monomial; making that step explicit would strengthen the write-up.