Conjugacy of finite biprefix codes

Two languages X and Y are called conjugates, if they satisfy the conjugacy equation XZ=ZY for some non-empty language Z. We will compare solutions of this equation with those of the corresponding equation of words and study the case of finite biprefix codes X and Y. We show that the maximal Z in this case is rational. We will also characterize X and Y in the case where they are both finite biprefix codes. This yields the decidability of the conjugacy of two finite biprefix codes.