Smooth normal forms of vector fields (the path method)

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?

• Have look at Brjuno's paper, Normal form of differential equations with a small parameter. (Russian) Mat. Zametki 16 (1974), 407–414 and also his famous work, Analytic form of differential equations. I, II. (Russian) Trudy Moskov. Mat. Obšč. 25 (1971), 119–262; ibid. 26 (1972), 199–239. – Liviu Nicolaescu Aug 5 '14 at 19:05