The present study aims at providing a complete picture of the various propagation scenarios that a statistically stationary turbulent premixed flame may possibly undergo. By explicitly splitting the scalar turbulent flux between its gradient and counter-gradient contributions, the scalar governing equation is rewritten as an ordinary differential equation in the phase space. Then, an analysis of the characteristic equations in the vicinity of the reactants and products side is carried out. The domain of existence of the propagation velocity is then determined and positioned over the relevant Bray number range. It is shown in particular that when a counter-gradient transport at the cold leading edge of the flame is dominant, there still exists a possibility of observing a steady regime of propagation. This conclusion is compatible with recent experimental data and observations based on the analysis of direct numerical simulations.