Fixed points of automorphisms preserving the length of words in free solvable groups

Let ?? be an automorphism of prime order p of the free group F _n . Suppose ?? has no fixed points and preserves the length of words. By ?? :=??? ^(m) we denote the automorphism of the free solvable group ${F_{n}/F_n^{(m)} }$ induced by ??. We show that every fixed point of ?? has the form ${cc^{\sigma} \ldots c^{\sigma^{p-1}}}$ , where ${c\in F_n^{(m-1)}/F_n^{(m)}}$ . This is a generalization of some known results, including the Macedo??ska?CSolitar Theorem 10].