A note on real forms of the complex N=4 supersymmetric Toda chain hierarchy in real N=2 and N=4 superspaces

Three inequivalent real forms of the complex N=4 supersymmetric Toda chain hierarchy (Nucl. Phys. B558 (1999) 545, solv-int/9907004) in the real N=2 superspace with one even and two odd real coordinates are presented. It is demonstrated that the first of them possesses a global N=4 supersymmetry, while the other two admit a twisted N=4 supersymmetry. A new superfield basis in which supersymmetry transformations are local is discussed and a manifest N=4 supersymmetric representation of the N=4 Toda chain in terms of a chiral and an anti-chiral N=4 superfield is constructed. Its relation to the complex N=4 supersymmetric KdV hierarchy is established. Darboux-Backlund symmetries and a new real form of this last hierarchy possessing a twisted N=4 supersymmetry are derived.