Singularities of 3–d vector fields preserving the form of Martinet

The paper works with germs at the origin and explicitly restricts μ-preserving conjugacies φ to those fixing the origin, which forces the associated 1D change ψ to be tangent to the identity (ψ′(0)=1) by φ(0,0)=(0,0) and (1+X)ψ′(y)=1+x (Lemma 2.3) . Consequently, the induced equivalence on univariate germs is the “restrictively 1–contact” relation g=(f∘ψ)/ψ′ with ψ′(0)=1 (Lemma 2.4 and Def. 3.1), under which the coefficient a in ay^k is an invariant for k≥1, and even constants are not rescalable across different a (Theorem 3.6) . The paper’s vector-field normal forms thus retain a modulus a in X1 and in Xk (k≥2), and it proves there are no hyperbolic singularities (Theorem 4.1) . By contrast, the model’s solution allows μ-preserving diffeomorphisms with g′(0)≠1, so φ_g(0,0)≠(0,0), which falls outside Diff(R^2,μ) as used in the paper. This larger group lets the model rescale away a in the cases f(0)≠0 and ord_0 f=k≥2, leading to the incorrect claims that all X0(a) (a≠0) and, for k≥2, all Xk(a) are μ-conjugate. Under the paper’s constraints, a is an invariant in k≥1, and even in the regular constant case a cannot be changed because ψ′(0)=1 enforces g(0)=f(0)=a (Lemmas 2.3–2.4, Theorems 3.6 and 4.1) .