We address the problem to know whether the relation induced by a one-rulelength-preserving rewrite system is rational. We partially answer to a conjecture of ÉricLilin who conjectured in 1991 that a one-rule length-preserving rewrite system is arational transduction if and only if the left-hand side u and theright-hand side v of the rule of the system are not quasi-conjugate orare equal, that means if u and v are distinct, there donot exist words x, y and z such thatu = xyz and v = zyx.We prove the only if part of this conjecture and identify two non trivialcases where the if part is satisfied.
