A NEW FAMILY OF INTEGRABLE DIFFERENTIAL SYSTEMS IN ARBITRARY DIMENSION

The paper states and proves Theorem 1 correctly: (i) H=I∘Φ is a first integral of ẋ=F via the chain rule and the adjugate identity, and non-constancy follows from the “preserves dimension” hypothesis; (ii) functional independence is preserved, using a contradiction and restricting to an open subset where det DΦ≠0; (iii) complete integrability near 0 when G(0)≠0 follows from the flowbox theorem (all directly in the paper’s proof of Theorem 1) . The candidate solution reproduces (i) and (iii), but (ii) hinges on a false claim: from “Φ preserves dimension” it concludes that the noncritical set U_*={det DΦ≠0} has full measure. The paper never asserts this, and it is not implied by the definition of preserving dimension; it only guarantees U_* is dense (otherwise there would exist an open set of critical points contradicting the definition). The paper’s proof avoids this pitfall by shrinking to an open subset where det DΦ≠0 to derive a direct contradiction with the independence of {∇I_j} on a full-measure set . Hence the model’s argument contains a critical gap, while the paper’s argument is sound.