I start by considering a polynomial vector field $$F=\varepsilon\frac{\partial}{\partial x}-(z^2+x)\frac{\partial}{\partial z}+0\frac{\partial}{\partial \varepsilon}.$$ Next I define a perturbation of $F$ as a vector field $X=F+\varepsilon G$.
I have been able to prove that there exists a formal change of coordinates $\hat\Phi$ such that $$ \hat\Phi_*( X)=F.$$
By Borel's lemma, there exists a smooth change of coordinates $\Phi$ such that $$ \Phi_*(X)=F+\tau(x,z,\varepsilon),$$ where $\tau$ is a smooth vector field on $\mathbb {R}^3$ whose components are flat at the origin, that is all partial derivatives of the components of $\tau$ at the origin are zero.
- I want to prove that there exists a smooth transformation $\Psi$ such that $\Psi_*(X)=F$. For this I tried to use the "path method", which consist on building a path between the vector fields $F$ and $F+\tau$. Such a path is an $s$-parameter family of vector fields $F_s=F+s\tau$. Next, we define the vector fields $$\tilde F=F_s+0\frac{\partial}{\partial s}$$ and $$ H=h_1\frac{\partial}{\partial x}+h_2\frac{\partial}{\partial z}+h_3\frac{\partial}{\partial \varepsilon} +1\frac{\partial}{\partial s}.$$ In this book there is a lemma stating that if $$[\tilde F,H]=0$$ then all $F_s$ are holomorphically equivalent to each other. I believe this is true also in the smooth case as the proof does not make any use of analyticity.
The problem is that I cannot find an explicit vector field $H$ satisfying the conditions of the lemma (this does not prove of course that $H$ does not exists, just that I am unable to find it) ... I can write down the computations that I have so far, if anyone wants.
So, is there any other way to do such things as conjugating two vector fields?