  We study the generalization of second Gelfand-Dickey bracket to the
superdifferential operators with matrix-valued coefficients. The associated
Miura transformation is derived. Using this bracket we work out a nonlocal and
nonlinear N=2 superalgebra which contains the N=2 super Virasoro algebra as a
subalgebra. The bosonic limit of this algebra is considered. We show that when
the spin-1 fields in this bosonic algebra are set to zero the resulting Dirac
bracket gives precisely the recently derived $V_{2,2}$ algebra.
