  We give an affrmative answer to the question whether a residually finite
Engel group satisfying an identity is locally nilpotent. More generally, for a
residually finite group G with an identity, we prove that the set of right
Engel elements of G is contained in the Hirsch-Plotkin radical of G. Given an
arbitrary word w, we also show that the class of all groups G in which the
w-values are right n-Engel and w(G) is locally nilpotent is a variety.
