N=2 Super Boussinesq Hierarchy: Lax Pairs and Conservation Laws

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-$W_3$ 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 $N=2$ pseudo-differential operators, while for $\alpha = -2$ the associated equation turns out to be bi-hamiltonian.