  We consider the integrable (zero perturbation) two--dimensional standard map,
in light of current developments on ergodic sums of irrational rotations, and
recent numerical evidence that it might possess non-trivial q-Gaussian
statistics. Using both classical and recent results, we show that the phase
average of the sum of centered positions of an orbit, for long times and after
normalization, obeys the Cauchy distribution (a q-Gaussian with q=2), while for
almost all individual orbits such a sum does not obey any distribution at all.
We discuss the question of existence of distributions for KAM tori.
