Orbits in free algebras of rank two

Let F be an algebra which is free in a class of algebras. In this paper we consider the following problem: If an endomorphism ψ of F preserves the orbit. (AutF)(w) of a nontrivial clement w ∈ F, does this imply that ψ is an automorphism? We give positive solutions of this problem when F is generated by two elements and is the free Lie algebra, free metabelian Lie algebra or free metabelian associative algebra, and w is a certain kind of element.link_to_subscribed_fulltex