We consider a two-wheeled single axle vehicle freely moving on the horizontal plane. This system is underactuated, nonholonomic and unstable. In order to achieve a simple control structure, the system is modeled on two levels: a nonholonomic kinematic model covering the motion of the axles midpoint and a holonomic kinetic model reflecting the (unstable) dynamics of the 3D multibody system. The kinematic model can be controlled via dynamic extension and exact feedback linearization while the multibody system is stabilized by a linear static state feedback. Both models are linked by the velocities of the wheels. The unavoidable lack of controllability for vanishing velocity is also addressed.