In 1959, Lyndon showed that in a free group, the equation $u^2v^2=w^2$ has only commuting solutions: $uv=vu=w$. Is there in the litterature any information about the following "twisted" version of the above equation: $f(u)ug(v)v={fg}(w)w$, where $f$ and $g$ are (commuting) automorphisms of the free group? Any indication welcome.