  Observational and numerical studies gave hints that the hot gaseous haloes of
ETGs may be sensitive to the galaxy internal kinematics. By using high
resolution 2D hydro simulations, and realistic two-component (stars plus dark
matter) axisymmetric galaxy models, we study the evolution of the hot haloes in
a suite of flat ETGs of fixed mass distribution, but with variable amounts of
azimuthal velocity dispersion and rotational support, including the possibility
of a counter-rotating inner stellar disc. The hot halo is fed by stellar mass
losses and heated by SNIa explosions and thermalization of stellar motions. We
measure the value of the thermalization parameter gamma (the ratio between the
heating due to the relative velocity between the stellar streaming and the ISM
bulk flow, and the heating attainable by complete thermalization of the stellar
streaming motions). We find that 1) the X-ray emission and the average
temperature are larger in fully velocity dispersion supported systems; 2)
0.1<gamma<0.2 for isotropic rotators (with a trend for being larger for lower
dark mass models); 3) systems that are isotropic rotators at large radii with
an inner counter-rotating disc, or fully velocity dispersion supported systems
with an inner rotating disc, have gamma=1, again with a trend to increase for
lower dark mass contents. We also find that the lower X-ray luminosities of
isotropic rotators cannot be explained just by their low gamma, but are due to
the complicated flow structure, consequence of the angular momentum stored at
large radii. X-ray emission weighted temperatures and luminosities nicely match
observed values; the X-ray isophotes are boxy in case of significant galaxy
rotation. Overall, it is found that rotation has an important role to explain
the observational result that more rotationally supported ETGs on average show
a lower X-ray emission [abridged].
