Analytic varieties invariant by foliations and Pfaff systems

The paper’s theorem (Corrêa, Theorem 6.1) proves that a codimension-r Pfaff system with at least dim H^1(X, Ω^1_cl) + dim(H^0(Ω^{r+1}⊗L)/(ω∧H^0(Ω^1_cl))) + r + 1 invariant hypersurfaces admits a meromorphic first integral, via a construction of r+1 global closed meromorphic 1-forms annihilated by wedge-with-ω and a two-case algebraic argument to extract a first integral. The model reproduces the same cohomological and residue framework and dimension count, but finishes with a kernel-bundle/K argument instead of the paper’s wedge-case split. Both are sound; the model’s proof is a slightly different but standard linear-algebraic route. Minor details the model skims (e.g., ensuring some coefficient is nonconstant) can be patched by the paper’s Lemma 6.2 step.