(see also about solvable algebras and solvable structures)
Let $M$ be a manifold and $X_1,\ldots,X_r$ vector fields on $M$. The condition
$$ [X_i,X_j]=\sum f_{ij}^k \cdot X_k, \tag{1} $$where $f_{ij}^k$ are functions, implies that the distribution $\mathcal{S}(\{X_1,\ldots,X_r\})$ generated by them is involutive. Frobenius theorem says that there exist integral manifolds.
But there is a more restricted condition, constant coefficients:
$$ [X_i,X_j]=\sum c_{ij}^k \cdot X_k \tag{2} $$where $c_{ij}^k \in \mathbb{R}$. This means that they constitute a finite dimensional Lie subalgebra of $\mathfrak{X}(M)$ (with (1) they constitute a possibly infinite dimensional one). In this case there exists a finite dimensional Lie group $G$ acting on $M$ such that the integral manifolds of the distribution $\mathcal{S}(\{X_1,\ldots,X_r\})$ are the orbits of $G$!!!
And even more, if every
$$ [X_i,X_j]=0 $$then the group is isomorphic to the translations $\mathbb{R}^r$.
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Author of the notes: Antonio J. Pan-Collantes
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