  Galactic globular clusters (GC) are known to have multiple stellar
populations and be characterised by similar chemical features, e.g. O-Na
anti-correlation. While second-population stars, identified by their Na
overabundance, have been found from the main sequence turn-off up to the tip of
the red giant branch in various Galactic GCs, asymptotic giant branch (AGB)
stars have rarely been targeted. The recent finding that NGC 6752 lacks an
Na-rich AGB star has thus triggered new studies on AGB stars in GCs, since this
result questions our basic understanding of GC formation and stellar evolution
theory. In order to compare the Na abundance distributions of AGB and RGB stars
in Galactic GCs and investigate whether the presence of Na-rich stars on the
AGB is metallicity-dependent, we obtained the high-resolution spectra with the
multi-object high-resolution spectrograph FLAMES on ESO/VLT for a sample of AGB
and RGB stars in the Galactic GC NGC 2808. The accurate Na abundances were
derived for 31 AGB and 40 RGB stars. We find that NGC 2808 has a mean
metallicity of -1.11 $\pm$ 0.08 dex, in good agreement with earlier analyses.
Comparable Na abundance dispersions are derived for our AGB and RGB samples,
with the AGB stars being slightly more concentrated than the RGB stars. The
ratios of Na-poor first-population to Na-rich second-population stars are 45:55
in the AGB sample and 48:52 in the RGB sample. NGC 2808 has Na-rich
second-population AGB stars, which turn out to be even more numerous - in
relative terms - than their Na-poor AGB counterparts and the Na-rich stars on
the RGB. Our findings are well reproduced by the fast rotating massive stars
scenario and they do not contradict the recent results that there is not an
Na-rich AGB star in NGC 6752. NGC 2808 thus joins the larger group of Galactic
GCs for which Na-rich second-population stars on the AGB have recently been
found.
