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