The first two Hamiltonian structures and the recursion operator connec
ting all evolution systems and Hamiltonian structures of the N = 2 sup
ersymmetric (n, m)-GNLS hierarchy are constructed in terms of N = 2 su
perfields in two different superfield bases with local evolution equat
ions. Their bosonic limits are studied in detail. New local and nonloc
al bosonic and fermionic integrals both for the N = 2 supersymmetric (
n, m)-GNLS hierarchy and its bosonic counterparts are derived. As an e
xample, in the n = 1, m = 1 case, the algebra and the symmetry transfo
rmations for some of them are worked out and a rich N = 4 supersymmetr
y structure is uncovered. (C) 1997 Elsevier Science B.V.