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We study the integrability properties of the one-parameter family of N = 2 super Boussinesq equations obtained earlier by two of us (E.I. & S.K., Phys. Lett. B 291 (1992) 63) as a Hamiltonian flow on the N = 2 super-W3 algebra. We show that it admits nontrivial higher order conserved quantities and hence gives rise to integrable hierarchies only for three values of the involved parameter, alpha = -2, -1/2, 5/2. We find that for the case alpha = -1/2 there exists a Lax pair formulation in terms of local N = 32 pseudo-differential operators, while for alpha = -653 2 the associated equation turns out to be bi-Hamiltonian.
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