Odd-dimensional pfaffian equations: Reduction to the hypersurface of singular points

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Let ω be a local nonvanishing differential 1-form on a (2k+1)-dimensional manifold with structurally smooth hypersurface S of singular points (the points at which us A (dw)k vanishes). We prove that in the holomorphic, real-analytic, and C ∞ categories, the Pfaffian equation (ω) is determined, up to a diffeomorphism, by its restriction to S, a canonical connection, and (in the real-analytic and the C ∞ cases) a canonical orientation. On the other hand, if we exclude certain degenerations of infinite codimension, then the restriction determines the connection and the orientation. Then (ω) is determined, up to a diffeomorphism, by its restriction to S.