Let us consider spherical coordinates $(r,\theta,\phi)$ on $\mathbb R^3$ and the manifold $M:=\mathbb R^3 \setminus \{0\}$.
Let us consider the 1-form on $M$ $$ \omega = zdz -r\sin^2{\theta}dr-\frac{1}{2}r^2\sin{2\theta}d\theta $$ where $z=r\cos{\theta}$. I think that we can write $\omega$ using only the spherical coordinates as $$ \omega = r\cos{2\theta}dr-r^2\sin{2\theta}d\theta $$ which is closed, $d\omega=0$. The ideal generated by $\omega$ is hence closed with respect to $d$, since $d(\eta \wedge \omega) = d\eta \wedge \omega$. This implies that the distribution $\Delta\subset TM$ (associated to $\omega$) is involutive, hence - by Frobenius theorem - integrable.
I have three questions:
1. I want to find the rank of the distribution and local generators of it: I think that the distribution is 2-dimensional and it is generated by $$ X = \frac{\partial}{\partial \phi} \qquad \text{and} \qquad Y=r^2\sin{2\theta}\frac{\partial}{\partial r}+r\cos{2\theta}\frac{\partial}{\partial\theta}. $$ Is it correct?
2. I want to find an integral variety of this distribution: I have found that the family of surfaces given by $$ \frac{r^2}{2}\cos{2\theta}-\phi+c=0, \qquad c \in \mathbb R. $$ are integral varieties of $\Delta$. Do you agree?
3. Finally, is $\Delta$ a trivial bundle over $M$? I do not know how I can do in order to answer this question. I have to establish if $\Delta \cong M \times \mathbb R$ (as vector bundles). Would you please help me?
Thanks.