In this presentation, we study the behavior of the Palais-Smale sequences of the conformal Dirac-Einstein equation, in 3-dimensional compact Riemannian manifolds. The problem originates from the super-symmetric model of coupling gravity with fermionic energy. Since the energy functional is critical, bubbling occurs. We give here a precise characterization of the violation of the (PS) condition and provide an Aubin type inequality guarantying the existence of solutions. Along the proof we see that this problem is in tight relation with the classical Yamabe problem and the Spinorial version of the Yamabe problem