On geometry of control systems equivalent to canonical contact systems: regular points, singular points, and flatness

We give necessary and sufficient geometric conditions for a nonlinear control system to be feedback equivalent to a canonical contact system for curves. We study the geometry of that class of systems, in particular, the existence of involutive subdistributions of corank one. We also distinguish regular points, at which the system is equivalent to the canonical contact system, and singular points. We show how the geometry of the studied class of systems is reflected by their flatness