Normal forms for the endpoint map near nice singular curves for rank two distributions

Given a rank-two sub-Riemannian structure (M, Delta) and a point x(0)is an element of M, a singular curve is a critical point of the endpoint map F:gamma?gamma (1) defined on the space of horizontal curves starting at x(0). The typical least degenerate singular curves of these structures are called regular singular curves; they are nice if their endpoint is not conjugate along gamma. The main goal of this paper is to show that locally around a nice singular curve gamma, once we choose a suitable topology on the control space we can find a normal form for the endpoint map, in which F writes essentially as a sum of a linear map and a quadratic form. This is a preparation for a forthcoming generalization of the Morse theory to rank-two sub-Riemannian structures.