The N = 2 super W 4 algebra and its associated generalized KdV hierarchies

We construct the N = 2 super W4 algebra as a certain reduction of the second Gel’fand-Dikii bracket on the dual of the Lie superalgebra of N = 1 super pseudodifferential operators. The algebra is put in manifestly N = 2 supersymmetric form in terms of three N = 2 superfields Φi(X), with Φ1 being the N = 2 energy momentum tensor and Φ2 and Φ3 being conformal spin 2 and 3 superfields respectively. A search for integrable hierarchies of the generalized KdV variety with this algebra as Hamiltonian structure gives three solutions, exactly the same number as for the W2 (super KdV) and W3 (super Boussinesq) cases.