Let $\mathcal{Z}=\mathcal{S}(Z_1,\ldots,Z_r)$ be an involutive distribution on a $n$-dimensional smooth manifold $M$. Let $\langle X_1,\ldots, X_{n-r}\rangle$ be an ordered collection of independent vector fields and define
$$ \mathcal{X}_k=\{ Z_1,\ldots,Z_r, X_1,\ldots,X_k\} $$for $k=1,\ldots,n-r$; and $\mathcal{X}_0=\{ Z_1,\ldots,Z_r\}$. We say that $\langle X_1,\ldots, X_{n-r}\rangle$ is a solvable structure for $\mathcal{Z}$ if:
1. $\mathcal{X}_k$ is a set of independent vectors for any $p\in M$ and for every $k=1,\ldots,n-r$.
2. The vector field $X_k$ is a symmetry of the rank $k+r$ distribution $\mathcal{S}(\mathcal{X}_k)$, for every $k=1,\ldots,n-r$.
Si en vez de tomar symmetry of a distribution hubiésemos pedido que fuesen cinf-symmetry of distribution habríamos obtenido una cinf-structure.
Su existencia permite obtener las inetral submanifolds de $\mathcal{Z}$ mediante $n$ cuadraturas.
Se puede debilitar la definición a una partial solvable structure
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Author of the notes: Antonio J. Pan-Collantes
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