Modules of vector fields, differential forms and degenerations of differential systems

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We deal with (n−1)-generated modules of smooth (analytic, holomorphic) vector fieldsV=(X1,..., Xn−1) (codimension 1 differential systems) defined locally on ℝn or ℂn, and extend the standard duality(X1,..., Xn−1)↦(ω), ω=Ω(X1,...,Xn−1,.,) (Ω−a volume form) betweenV′s and 1-generated modules of differential 1-forms (Pfaffian equations)—when the generatorsXi are linearly independent—onto substantially wider classes of codimension 1 differential systems. We prove that two codimension 1 differential systemsV and\((\tilde X_1 ,...,\tilde X_{n - 1} )\) are equivalent if and only if so are the corresponding Pfaffian equations (ω) and\((\tilde \omega )\) provided that ω has1-division property: ωΛμ=0, μ—any 1-form ⇒ μ=fω for certain function germf. The 1-division property of ω turns out to be equivalent to the following properties ofV: (a)fX∈V, f—not a 0-divisor function germ ⇒X∈V (thedivision property); (b) (V⊥)⊥=V; (c)V⊥=(ω); (d) (ω)⊥=V, where ⊥ denotes the passing from a module (of vector fields or differential 1-forms) to its annihilator.