Do Flat Systems Admit a Flat Output Compatible with a Symmetry Group Action?

We study the problem posed by Martin et al. whether a flat control system, invariant by a symmetry group action, possesses always a flat output compatible with that action. For driftless systems with 3 states and 2 controls and any symmetry group we characterize all flat outputs compatible with its action. We show that 1-dimensional groups with no fixed points and strong symmetry groups always possess compatible flat outputs. We construct an action of a 4-dimensional group for which no compatible flat output exists thus providing a negative answer to the question of [7], [10].