It is well known that the image of a free Lie algebra action $\rho:\mathfrak{g}\to\mathfrak{X}^1(M)$ on a manifold $M$ is an integrable distribution of constant rank $(=\rm{dim}\;\mathfrak{g})$. Thus it defines a foliation $\mathcal{F}$ on $M$. The leaf of $\mathcal{F}$ are called the $\mathfrak{g}$orbits of the action. Conversely, given a foliation $\mathcal{F}$ on a manifold $M$, what are the conditions on $\mathcal{F}$ to be the foliation of a (free) Lie algebra action $\rho:\mathfrak{g}\to\mathfrak{X}^1(M)$, i.e., the leaves of $\mathcal{F}$ are the $\mathfrak{g}$orbits of $\rho$ ?
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If i remember correctly the subject was somewhat fashionable in the 60ies and 70ies, when some papers on the subject appeared. I list a couple of references which can give an idea about what was obtained at the time and that can help you in further inquiries: E. Lima, commuting vector fields on $\mathbb S^3$, Ann. Math. 81, 7081 1965. Pasternack, Foliations and compact Lie group actions, Comment. Math. Helv. 46, 467477 1971. 

