  Using the methods of the theory of formal symmetries, we obtain new easily
verifiable sufficient conditions for a recursion operator to produce a
hierarchy of local generalized symmetries. An important advantage of our
approach is that under certain mild assumptions it allows to bypass the
cumbersome check of hereditariness of the recursion operator in question, what
is particularly useful for the study of symmetries of newly discovered
integrable systems. What is more, unlike the earlier work, the homogeneity of
recursion operators and symmetries under a scaling is not assumed as well. An
example of nonhereditary recursion operator generating a hierarchy of local
symmetries is presented.
