Take $M=\mathbb{R}^3$ be a smooth manifold. Consider a distribution
$\Delta_{(x,y,z)} = Span\{y\frac{\partial}{\partial x}-x\frac{\partial}{\partial y}, z\frac{\partial}{\partial x} - x\frac{\partial}{\partial z} \}$.
1) Show that the distribution is integrable. 2) Describe the maximal integral submanifolds.
Here is some of my drafts:
Take $V = y\frac{\partial}{\partial x}-x\frac{\partial}{\partial y}$ and $W = z\frac{\partial}{\partial x} - x\frac{\partial}{\partial z}$, then the Lie bracket $[V,W]=VW-WV=-y\frac{\partial}{\partial z}+z\frac{\partial}{\partial y}$.
If $[V_p, W_p]\in \Delta_p$, then the distribution is involutive and further integrable. However, for $p=(0,y,z)$, $V_p=y\frac{\partial}{\partial x}$, $W_p = z\frac{\partial}{\partial x}$, and $[V_p, W_p]=-y\frac{\partial}{\partial z}+z\frac{\partial}{\partial y}$. Easy to see the Lie bracket is not in the span. If this is true, then the distribution is not integrable. Could anyone remind me what is wrong?
As to the second part, I did not have a clear clue of how to complete. A concrete computation would be appreciated.