Integral manifold I'm solving the following:
Let $D$ be a distribution on $\mathbb{R}^3/\{(0,y,z):y,z\in\mathbb{R}\}$ with basis vector fields $X=z\frac{\partial}{\partial x}-x\frac{\partial}{\partial z}$
$Y=-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y}$. Find images of integral manifolds of distribution $D$.
From $x'=z$ and $z'=-x$, I get $xx'+zz'=0$ and $x^2+z^2=const$. Similarly, from equation for $Y$, $x^2+y^2=const$. 
Can I now conclude that the wanted integral manifold is $x^2+y^2+z^2=const$ (since $X(x^2+y^2+z^2-const)=0$ and $Y(x^2+y^2+z^2-const)=0$)? If my conclusion isn't correct, how to find exact solution?
Thanks in advance.

The distribution is annihilated by the $1$-form $\omega=dx+\dfrac1x\big(y\,dy+z\,dz\big)$. Since $x\omega=d\big(\frac12(x^2+y^2+z^2)\big)$, integral manifolds are in fact spheres centered at the origin.
