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A 1-plane field on a manifold is a smooth choice of a 1 dimensional subspace of the tangent space at every point. If this plane field is integrable then there is an associated 1-dimensional foliation on the manifold that has the plane field as its tangent space. Parameterizing the leaves of the foliation would seem to give a one parameter group of diffeomorphisms. Likewise, to every one parameter group of diffeomorphisms there is an associated vector field (which could be thought of as a smooth 1-plane field), and at any point flowing as long as one could in both directions would seem to give a leaf. The union of these leaves would then seem to give a foliation.

Can someone confirm if this line of reasoning is essentially correct. These ideas seem to be the same except for foliations don't come with a parameterization. I'm guessing there is a natural way to parameterize using the charts though.

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The leaves may not be coherently orientable. So your plane field (really a line field) may not lift to a vector field, let alone a 1-parameter family of diffeomorphisms. You can construct non-orientable line fields in $\mathbb R^2 \setminus \{0\}$.

Provided your line field has no orientability obstruction, it lifts to a vector field. Vector fields don't always integrate to 1-parameter families of diffeomorphisms for completeness issues -- but in various circumstances like if your manifold was compact, then it would have to integrate.

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