Let $D$ be the distribution on $M=\{(x,y,z),x,y,z>0\}$ generated by $X=y\frac{\partial}{\partial z}-z\frac{\partial}{\partial y}$ and $Y=z\frac{\partial}{\partial x}-x\frac{\partial}{\partial z}$, show $D$ is involutive and find the general integral manifolds of $D$.
I compute the flows of $X$ and $Y$, there are a lot of $\sin$ and $\cos$ in the form of flows of $X$ and $Y$ , I got $\phi_t(x,y,z)=(x,-z\sin t+y\cos t,y\sin t+z\cos t)$ and $\psi_s(x,y,z)=(z\sin s+x\cos s,y,-x\sin s+z\cos s)$, but to find the integral manifolds at last, I can not eliminate $s$ and $t$ to get a general form
Best Answer
Differential forms for the win. Note that if you take the $1$-form $$\omega = x\,dx+y\,dy+z\,dz,$$ then $\omega(X)=\omega(Y)=0$. So the distribution is given by the kernel of $\omega$. Of course, $\omega = d\big(\frac12(x^2+y^2+z^2)\big)$, so integral manifolds of $D$ are the level sets $x^2+y^2+z^2=c$ for $c>0$.