N = 4 super NLS-mKdV hierarchies

N = 2 extension of affine algebra sl(2) ⊕ u(1) possesses a hidden global N = 4 supersymmetry and provides a second hamiltonian structure for a new N = 4 supersymmetric integrable hierarchy defined on N = 2 affine supercurrents. This system is an N = 4 extension of, at the 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 construction which relates N = 2 sl(2) ⊕ u(1) algebra to “small” N = 4 SCA. We also find the underlying affine hierarchy for another integrable system with the N = 4 SCA second hamiltonian structure, “quasi” N = 4 KdV hierarchy. It respects only N = 2 supersymmetry. For both new hierarchies we construct scalar Lax formulations. We speculate that any N = 2 affine algebra admitting a quaternionic structure possesses N = 4 supersymmetry and so can be used to produce N = 4 supersymmetric hierarchies. This suggests a way of classifying all such hierarchies.