  Let m, n be positive integers, v a multilinear commutator word and w = v^m.
Denote by v(G) and w(G) the verbal subgroups of a group G corresponding to v
and w, respectively. We prove that the class of all groups G in which the
w-values are n-Engel and w(G) is locally nilpotent is a variety (Theorem A).
Further, we show that in the case where m is a prime-power the class of all
groups G in which the w-values are n-Engel and v(G) has an ascending normal
series whose quotients are either locally soluble or locally finite is a
variety (Theorem B). We examine the question whether the latter result remains
valid with m allowed to be an arbitrary positive integer. In this direction, we
show that if m, n are positive integers, u a multilinear commutator word and v
the product of 896 u-words, then the class of all groups G in which the
v^m-values are n-Engel and the verbal subgroup u(G) has an ascending normal
series whose quotients are either locally soluble or locally finite is a
variety (Theorem C).
