In this paper we investigate analytic affine control systems \hbox{$\dot{q}$}q̇
= X + uY, u ∈ [a,b] , where
X,Y is an orthonormal frame for a generalized Martinet sub-Lorentzian
structure of order k of Hamiltonian type. We construct normal forms for
such systems and, among other things, we study the connection between the presence of the
singular trajectory starting at q0 on the boundary of the
reachable set from q0 with the minimal number of analytic
functions needed for describing the reachable set from q0.