This is a small problem which drives me crazy.
Let $\varphi(x,y,t)=(F_1(x,y,t),F_2(x,y,t))$ be a one paramter transformation group on $\mathbb{R}^2$.
Let $F_3(x,y,z,t)=\frac{D_1F_2+zD_2F_2}{D_1F_1+D_2F_1}$, then $\varphi^1(x,y,z,t)=(F_1, F_2, F_3)$ is also a one parameter group.
If the infinitesimal generator of $\varphi$ is $\xi\frac{\partial}{\partial x }+\eta\frac{\partial}{\partial y}$, prove that the infinitesimal generator of $\varphi^1$ is
$\xi\frac{\partial}{\partial x }+\eta\frac{\partial}{\partial y}+[\eta_x+z(\eta_y-\xi_x)-z^2\xi_y]\frac{\partial}{\partial z}$
I just can't get the right form. I hope someone can show me the proof with specific procedures.