On the class of non-linear control systems mapping onto linear systems

The control system in the form $\dot x=a(x)+b(x)u$, $x\in{\mathbf R}^n$, $u\in{\mathbf R}$, is considered. In the term of Lee brackets necessary and sufficient conditions of possibility to map of this control system (without change of a control) onto the system with additive control, in particular, onto the linear control system with respect to $x$ and $u$ are given. The conditions of local controllability of systems mapping onto linear system are formulated.