Feedback classification of nonlinear control systems on 3-manifolds

We consider nonlinear control-affine systems with two inputs evolving on three-dimensional manifolds. We study their local classification under static state feedback. Under the assumption that the control vector fields are independent we give complete classification of generic systems. We prove that out of a singular smooth curve a generic control system is either structurally stable and thus equivalent to one of six canonical forms (models) or finitely determined and thus equivalent to one of two canonical forms with real parameters. Moreover, we show that at points of the singular curve the system is not finitely determined and we give normal forms containing functional moduli. We also study geometry of singularities, i.e., we describe surfaces, curves, and isolated points where the system admits its canonical forms.