Stable elements in free nilpotent groups of rank two

The article presents a complete description of the elements of free nilpotent groups of rank two and stage 8 that remain unchanged under the action of any automorphism of the group. Existence of nontrivial stable elements in such groups is known, but their number was not found. We prove a theorem on the uniqueness (up to its powers) of nontrivial stable element in free nilpotent groups of rank two and stage 8.