Holomorphic Vector Fields with Real Integral Manifolds

The paper proves a complete dichotomy and explicit normal forms for all holomorphic vector fields in C^2 that admit a Levi-nonflat real-analytic integral 3-fold through the singularity. Its Theorem 1 lists the finite-type normal forms (1–6) and the infinite-type normal forms (7–13), plus a single purely formal, generically divergent exceptional case (14), and also proves realizability of every listed form by an analytic pair (X, M) . The introduction makes clear that this completes the classification of infinitesimal symmetries of Levi-nonflat hypersurfaces in C^2, including the previously open nonminimal (infinite-type) regime . The paper provides the normalization and convergence theorems in the infinite-type setting (e.g., Theorem 2 for the generic case and Theorem 3 for the exceptional case with α_k≡0) and a formal normal form with a divergence result for case (14) (Section 4) . By contrast, the candidate solution asserts that the infinite-type classification with explicit forms (7)–(14) “appears to have been open” as of 2024-08-09; this is directly contradicted by the paper’s main theorem and proofs. The finite-type portion of the candidate solution broadly agrees with the classical Chern–Moser/Kolář picture (and the paper’s summary of that background) , but the overall claim that the infinite-type classification was not available is incorrect.