  Classical Cepheids (DCEPs) are the most important primary indicators for the
extragalactic distance scale. Establishing the dependence on metallicity of
their period--luminosity and period--Wesenheit (PL/PW) relations has deep
consequences on the estimate of the Hubble constant (H$_0$). We aim at
investigating the dependence on metal abundance ([Fe/H]) of the PL/PW relations
for Galactic DCEPs. We combined proprietary and literature photometric and
spectroscopic data, gathering a total sample of 413 Galactic DCEPs (372
fundamental mode -- DCEP\_F and 41 first overtone -- DCEP\_1O) and constructed
new metallicity-dependent PL/PW relations in the near infra-red (NIR) adopting
the Astrometric Based Luminosity. We find indications that the slopes of the
PL$(K_S)$ and PW$(J,K_S)$ relations for Galactic DCEPs might depend on
metallicity when compared to the Large Magellanic Cloud relationships.
Therefore, we have used a generalized form of the PL/PW relations to
simultaneously take into account the metallicity dependence of the slope and
intercept of these relations. We calculated PL/PW relations which, for the
first time, explicitly include a metallicity dependence of both the slope and
intercept terms. Although the insufficient quality of the available data makes
our results not yet conclusive, they are relevant from a methodological point
of view. The new relations are linked to the geometric measurement of the
distance to the Large Magellanic Cloud and allowed us to estimate a {\it Gaia}
DR2 parallax zero point offset $\Delta \varpi$=0.0615$\pm$0.004 mas from the
dataset of DCEPs used in this work.
