Assume that $X$ is a non-vanishing vector field on $\mathbb{R}^3$.



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> Is there a $2$-dimensional foliation of space such that every trajectory of $X$ is contained in a leaf of the $2$-dimensional foliation?
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As a related question:



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> Is there a classification of all $1$-dimensional foliations of space tangent to the unit speed vector field $t$ for which the distributions $\{t,n\}$ and $\{t,b\}$ are integrable? Here $\{t,n,b\}$ is the associated Frenet frame.
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$2$ dimensional foliations of space whose leaves contain the trajectories of a given vector field
No. For a counterexample, start with the Hopf map $S^3\to S^2$, a fiber bundle with $S^1$ fibers. Its fibers are the leaves of a $1$-dimensional foliation of $S^3$ in which all leaves are closed and the space of leaves is $S^2$. Choose a vector field tangent to the leaves. Remove one point from $S^3$ to get $\mathbb R^3$. A $2$-dimensional foliation of the kind asked for would give a $1$-dimensional foliation of $S^2$.


