Consider the control system with state $q=(x,y,z)\in \mathbb{R}^3$ defined by $$ \dot q = u_1F_1(q)+u_2F_2(q) $$ where $u_1,u_2$ are the control functions (measurable mappings) and $F_1,F_2$ are the vector fields defined by $$ F_1 = \partial_x + (-15\,{x}^{2}y-5\,{y}^{3}+{x}^{2}-2\,yx+{y}^{2}+12\,y)\partial_z $$ $$ F_2 = \partial_y $$
Using the following actions on couple (F,G) of vector fields:
- action of a diffeomorphism $\varphi$ (change of coordinates): $\varphi \cdot (F,G) = (\varphi_\ast(F), \varphi_\ast(G))$
- action of a formal power series $\alpha$ in x,y,z : $\alpha \cdot (F,G) = (\cos(\alpha) F +\sin(\alpha) G,-\sin(\alpha) F +\cos(\alpha) G)$,
I would like to transform the system into a new one where the new vector fields $\hat F_1,\hat F_2$ can be expressed as $$ \hat F_1 = \partial_x + r(x,y)\partial_z $$ $$ \hat F_2 = \partial_y +s(x,y) \partial_z $$ where $r$ and $s$ are polynomials functions which doesn't contain any of the second order terms $x^2,y^2,xy$. Moreover the transformation should let invariant $(0,0,0)$.
EDIT: In page 21 of the following article,
http://sci-hub.cc/10.1007/978-1-4612-5651-9_2
the author gave explicit transformation to remove those terms but it seems there is an error of computations.