  Spectroscopic observations obtained with the VLT of one planetary nebula (PN)
in Sextans A and of five PNe in Sextans B and of several HII regions (HII) in
these two dwarf irregular galaxies are presented. The extended spectral
coverage, from 320.0 to 1000.0nm, and the large telescope aperture allowed us
to detect a number of emission lines, covering more than one ionization stage
for several elements (He, O, S, Ar). The electron temperature (Te) diagnostic
[OIII] line at 436.3 nm was measured in all six PNe and in several HII allowing
for an accurate determination of the ionic and total chemical abundances by
means of the Ionization Correction Factors method. For the time being, these
PNe are the farthest ones where such a direct measurement of the Te is
obtained. In addition, all PNe and HII were also modelled using the
photoionization code CLOUDY. The physico-chemical properties of PNe and HII are
presented and discussed. A small dispersion in the oxygen abundance of HII was
found in both galaxies: 12 + $\log$(O/H)=7.6$\pm$0.2 in SextansA, and
7.8$\pm$0.2 in SextansB. For the five PNe of SextansA, we find that 12 +
$\log$(O/H)=8.0$\pm$0.3, with a mean abundance consistent with that of HII. The
only PN known in SextansA appears to have been produced by a quite massive
progenitor, and has a significant nitrogen overabundance. In addition, its
oxygen abundance is 0.4 dex larger than the mean abundance of HII, possibly
indicating an efficient third dredge-up for massive, low-metallicity PN
progenitors. The metal enrichment of both galaxies is analyzed using these new
data.
