ON A THEOREM OF LYAPUNOV-POINCARÉ IN HIGHER DIMENSION

The paper’s Theorem A states that a real-analytic integrable 1-form whose first jet equals d(∑_{j=1}^r x_j^2) has a real-analytic first integral when either r=2 with closed leaves, or r≥3. The paper proves (i) by slicing to a generic 2-plane, obtaining a planar analytic first integral and then extending it holomorphically to the ambient via Mattei–Moussu; and proves (ii) by a slicing/induction scheme that reduces to Reeb’s theorem when r=n−1 and then extends from hyperplane slices (with the extension result of [8]) to the ambient dimension . The candidate solution handles (i) similarly (planar reduction and extension), but cites Dulac’s center theorem for the planar first integral; for (ii) it takes a different route, using a codimension estimate for Sing(Ω) and invoking Malgrange’s Frobenius-with-singularities theorem to get Ω = g dF directly in C^n, then restricting to R^n. Both approaches correctly yield the stated conclusions. Minor gaps in the candidate’s write-up (e.g., the need to assume the slice is in general position for the extension step, as the paper explicitly does, and to state the precise hypotheses of the Malgrange theorem used) do not invalidate the argument. The paper’s proof is complete and self-contained within its chosen toolkit; the model’s proof is a valid alternative using standard results.