  The V-algebras are the non-local matrix generalization of the well-known
W-algebras. Their classical realizations are given by the second Poisson
brackets associated with the matrix pseudodifferential operators. In this
paper, by using the general Miura transformation, we give the decomposition
theorems for the second Poisson brackets, from which we are able to construct
the free field realizations for a class of V-algebras including
V_{(2k,2)}-algebras that corresponds to the Lie algebra of C_k-type as the
particular examples. The reduction of our discussion to the scalar case
provides the similar result for the W_{BKP}-algebra.
