N=4 Toda chain (KdV) hierarchy in N=4 superspace

The Lax pair and Hamiltonian formulations are presented for the N=4 supersymmetric Toda chain (KdV) hierarchy in N=4 superspace. The general formulas for an infinite tower of its bosonic flows in terms of the Lax operator in N=4 superspace are derived, and five real forms of the hierarchy are presented. New N=4 superfield bases in which the flows are local are discussed. A relation between the two descriptions of the hierarchy in N=4 superspace used in the literature is established.