## Abstract

We study germs of holomorphic distributions with “separated variables”. In codimension one, a well know example of this kind of distribution is given by

which defines the canonical contact structure on \({\mathbb {C}}{\mathbb {P}}^{2m+1}\). Another example is the Darboux distribution

which gives the normal local form of any contact structure. Given a germ \({\mathcal {D}}\) of holomorphic distribution with separated variables in \(({\mathbb {C}}^n,0)\), we show that there exists , for some \(\kappa \in {\mathbb {Z}}_{\ge 0}\) related to the Taylor coefficients of \({\mathcal {D}}\), a holomorphic submersion

such that \({\mathcal {D}}\) is completely non-integrable on each level of \(H_{{\mathcal {D}}}\). Furthermore, we show that there exists a holomorphic vector field *Z* tangent to \({\mathcal {D}}\), such that each level of \(H_{{\mathcal {D}}}\) contains a leaf of *Z* that is somewhere dense in the level. In particular, the field of meromorphic first integrals of *Z* and that of \({\mathcal {D}}\) are the same. Between several other results, we show that the canonical contact structure on \({\mathbb {C}}{\mathbb {P}}^{2m+1}\) supports a Legendrian holomorphic foliation whose generic leaves are dense in \({\mathbb {C}}{\mathbb {P}}^{2m+1}\). So we obtain examples of injectively immersed Legendrian holomorphic open manifolds that are everywhere dense.

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Maycol Falla Luz was partially supported by CAPES-COFECUB Ma932/19. Rudy Rosas was supported by the Vicerrectorado the Investigación de la Pontificia Universidad Católica del Perú

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Luza, M.F., Rosas, R. Distributions, First Integrals and Legendrian Foliations.
*Bull Braz Math Soc, New Series* **53**, 1157–1229 (2022). https://doi.org/10.1007/s00574-022-00300-0

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DOI: https://doi.org/10.1007/s00574-022-00300-0