N=4 SUPER NLS-MKDV HIERARCHIES

N = 2 extension of affine algebra sl(2) circle plus u(1) possesses a h idden global N = 4 supersymmetry and provides a second hamiltonian str ucture for a new N = 4 supersymmetric integrable hierarchy defined on N = 2 affine supercurrents. This system is an N = 4 extension of, at t he same time, two hierarchies: N = 2 NLS and N = 2 mKdV. It is related to the N = 4 KdV hierarchy via a generalized Sugawara-Feigin-Fuks con struction which relates N = 2 sl(2) circle plus u(1) algebra to ''smal l'' N = 4 SCA. We also find the underlying affine hierarchy for anothe r integrable system with the N = 4 SCA second hamiltonian structure, ' 'quasi'' N = 4 KdV hierarchy. It respects only N = 2 supersymmetry. Fo r both new hierarchies we construct scalar Lax formulations. We specul ate that any N = 2 affine algebra admitting a quaternionic structure p ossesses N = 4 supersymmetry and so can be used to produce N = 4 super symmetric hierarchies. This suggests a way of classifying all such hie rarchies. (C) 1997 Elsevier Science B.V.