Asymptotic Lines and Parabolic Points of Plane Fields in R3

The paper’s Theorem 4.13 proves that for a C^k-generic vector field ξ in R^3, the orthogonal plane field has (i) a regular parabolic surface P, (ii) a regular curve of special parabolic points whose points are generically of saddle/node/focus type with isolated saddle–node, node–focus, and Hopf transitions, and (iii) all other parabolic points are cuspidal. This is established via a Lie–Cartan formulation and jet-transversality in Section 4 (Definition 4.12 and Theorem 4.13), with the equivalence of “special parabolic = singular of the Lie–Cartan field” proved in Lemma 2.54 and the local cusp model proved in Theorem 3.2. The candidate solution reaches the same conclusions using a closely related but somewhat different transversality setup (a submersion argument for H=(F,F_p,G) on the prolongation), and invokes standard BDE classification. Hence both are correct and consistent with the paper’s statements and proofs (Theorem 4.13; Lemma 2.54; Proposition 2.47; Theorem 3.2) .