A single potential impurity can drastically change a low-temperature transport of strongly interacting particles in one dimension - a sys- tem which is known as the Luttinger liquid. An arbitrary weak backscattering of fermions from the impurity totally destroys their zero-temperature current while even a very strong backscattering of bosons makes no impact on their flow. On the other hand, a more complicated impurity (like a quantum dot or a double-barrier struc- ture with a resonant level) can preserve an ideal resonant conductance of fermions, or conversely lead (in a different geometry) to an ideal (i.e. infinite) resonant resistance. I emphasize the role of a so-called duality in these transport effects. The duality was related to the integrability of the Luttinger liquid with an impurity. I will show that - surprisingly - duality survives the addition of a non-local and retarded interaction (like electron-phonon) which almost certainly destroys the integrability.