Consider the vector fields in $\Bbb{R}^2$:
$$F(x,y)=(py,-px),\quad G(x,y)=(qy,-qx) $$
I want to prove that these vector fields are conjugated $\iff |p|=|q|$
By definition, $F$ and $G$ are conjugate if exists a diffeomorphism $H$ such that
$$H\,\circ F_t = G_t \,\circ H, $$
where $F_t$ and $G_t$ are the trajectories of $F$ and $G$. Note that
$$F(x,y)=\begin{bmatrix}0 & p \\ -p & 0\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix},\quad G(x,y)=\begin{bmatrix}0 & q \\ -q & 0\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix} $$
So, the trajectories are given by the exponential of these matrix:
$$F_t(x,y)=\begin{bmatrix}\cos pt & \sin pt \\ -\sin pt & \cos pt\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix},\quad G_t(x,y)=\begin{bmatrix}\cos qt & \sin qt \\ -\sin qt & \cos qt\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix} $$
My first attempt is work with linear $H$, that is, $H(x,y)=C\begin{bmatrix}x \\ y\end{bmatrix}$ for some matrix $C$. So, I need to work with
$$C e^{At}=e^{Bt}C, $$
where $A$ and $B$ are the matrix in the definition of $F$ and $G$ respectively.
What can I do?