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The N=2 super W4 algebra is constructed 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 Phi i(X), with Phi1 being the N=2 energy momentum tensor and Phi2 and Phi3 being conformal spin 2 and 3 superfields, respectively. A search for integrable hierarchies of the generalized Korteweg-de Vries (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.
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