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Pia, Nicola (2021): On the dynamics of some vector fields tangent to non-integrable plane fields. In: Journal of Symplectic Geometry, Vol. 19, No. 2: pp. 399-412

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Let E-3 subset of TMn be a smooth 3-distribution on a smooth n-manifold, and W subset of E a line field such that [W, E] subset of E. We give a condition for the existence of a plane field D-2 such that W subset of D and [D, D] = E near a closed orbit of W. If W has a non-singular Morse-Smale section, we get a condition for the global existence of D. As a corollary we obtain conditions for a non-singular vector field W on a 3-manifold to be Legendrian, and for an even contact structure E subset of TM4 to be induced by an Engel structure D.

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