ON N

Using the FS and HST versions of the free N = 4 matter multiplet (O4,( 1/2)4), we construct two N = 4 SU(2) conformal superfield models. The corresponding N = 4 conserved currents are given. We find that no N = 4 SU(2) Liouville model exists as long as the SU(2) KM symmetry is man ifestly preserved. However allowing an explicit breaking of the SU(2) KM subsymmetry of the N = 4 conformal algebra down to U(1) KM, we obta in & Feigin-Fuchs extension of the N = 4 supercurrent showing that N = 4 Liouville theory and its Toda generalizations could exist. Quantiza tion and the N = 4 conformal anomaly are studied.