Orthogonal surfaces to a vector field

Question

Consider the vector field $\mathbf F(x,y,z)=(y-x,x-y-xz,xy-z)$.

1. Find the equations of the characteristic curves of $\mathbf F$ and discuss the existence of orthogonal surfaces to $\mathbf F$.
2. Find the family of curves of part 1 restricted to the plane $z=0$.

Regarding the first part, the characteristic curves satisfy

$$\frac{dx}{y-x}=\frac{dy}{x-y-xz}=\frac{dz}{xy-z}$$

or

$$\begin{align} (x-y-xz)dx&=(y-x)dy\\ (xy-z)dx&=(y-x)dx\\ (x-y-xz)dz&=(xy-z)dy \end{align}$$

and, if I am not mistaken, the existence of such surfaces is guaranteed locally since $\mathbf F$ is continuous.

However, I am not sure what I am supposed to do in part 2. Any help would be appreciated.

Answer 1

If a characteristic curve $p(t)=(x(t),y(t),0)$ stays inside the $z=0$ plane, then $\textbf{F}(x(t),y(t),0)$ should have its $z$ component stay $0$, so $$x(t)y(t)=0$$, and $$-(x-y)dy=(x-y)dx$$ Then solve these.