Make two independent vector fields commute by multiplying them on functions

Question

Let $\xi$ and $\eta$ be two independent vector fields on plane. Is it possible to make them commute by multiplying by some functions $f$ and $g$, i.e. $[f\xi,g\eta]=0$?

Answer 1

HINT:

You can do it at least locally. Consider the flow lines of $\chi$, and of $\eta$, around a point. The statement is equivalent to: the pair flow lines look locally like $y=$ const, and $x=$ const. That would also give you the coordinate system required. Of course, we can also calculate, but this is the idea.