Normalization of homogeneous approximations of symmetric affine control systems with two controls

In this paper, we study properties of growth vectors and homogeneous approximations of symmetric affine control systems. We say that a growth vector is A-normal if homogeneous approximations of all systems with this growth vector can be reduced (by use of feedbacks) to a finite number of "normal forms," and a growth vector is said to be A-simple if the set of homogeneous approximations of all small perturbations of systems with this growth vector satisfies this property. We give the complete description of A-normal and A-simple growth vectors for systems with two-dimensional controls. As the main tool, we develop the free algebraic technique.