Solvable structure

Let $\mathcal{Z}=\mathcal{S}(Z_1,\dots ,Z_r)$ be an involutive distribution on a $n$-dimensional smooth manifold $M$. Let $⟨X_1,\dots ,X_{n-r}⟩$ be an ordered collection of independent vector fields and define

for $k=1,\dots ,n-r$; and ${\mathcal{X}}_0=\{Z_1,\dots ,Z_r\}$. We say that $⟨X_1,\dots ,X_{n-r}⟩$ 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 $k=1,\dots ,n-r$.
2. The vector field $X_k$ is a symmetry of the rank $k+r$ distribution $\mathcal{S}({\mathcal{X}}_k)$, for every $k=1,\dots ,n-r$.

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