I have the following differential equation:
${\dot{z}} = -iz + \epsilon [\frac{z-z*}{2} + \frac{(z-z*)^2}{24}]$
(the parameter $z$ is complex and i is the classic imaginary $\sqrt{-1}$)
I'm trying to use a Lie transformation in order to find the normal form of this equation approximated through O($\epsilon^2$), in order to get a simpler version of my original equation (simplifies my Physical model).
I'm having trouble with the Lie transformations (of the kind $z = e^{\epsilon G}w$) because I'm self-taught on the matter. Can someone help me with the specific problem, or maybe guide me on how to do it myself?