  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.
