  The massless boundary sine-Gordon (SG) model is the only interacting impurity
model with a known exact solution out-of-equilibrium, yet existing so far only
for integer values of the sine Gordon coupling $\lambda$ [Phys. Rev. Lett.
{\bf74}, 3005 (1995)]. We present here a full exact solution for arbitrary
rational values of $\lambda$, at arbitrary voltage $V$ and temperature $T$. We
use the "string" solutions of the bulk SG model, here regarded as genuine
quasiparticles avoiding charge diffusion in momentum space. We carefully
present the finite voltage and temperature thermodynamics of this gas of
interacting exotic quasiparticles, whose very nature depends on subtle
arithmetic properties of the rational SG parameter $\lambda$, and explicitly
check that the string representation is thermodynamically complete. By
considering a Loschmidt echo, we derive the exact transmission probability of
strings on the impurity. We obtain the exact universal scaling function for the
electrical current $I(V,T)$. Our results are in excellent agreement with recent
experimental out-of-equilibrium data and question the reality of these exotic
quasiparticles.
