Simple equations on binary factorial languages

We consider equations on the monoid of factorial languages on the binary alphabet. We use the notion of a canonical decomposition of a factorial language and previous results by Avgustinovich and the author to solve several simple equations on binary factorial languages including Xn=Yn, the commutation equation XY=YX and the conjugacy equation XZ=ZY. At the end of the paper, we discuss the difficulties hindering the reduction of equations on factorial languages to equations on words and the extension of the alphabet considered.