  Anomalous Cepheids (ACEPs) are intermediate mass metal-poor pulsators mostly
discovered in dwarf galaxies of the Local Group. However, recent Galactic
surveys, including the Gaia DR3, found a few hundreds of ACEPs in the Milky
Way. Their origin is not well understood. We aim to investigate the origin and
evolution of Galactic ACEPs by studying for the first time the chemical
composition of their atmospheres. We used UVES@VLT to obtain high-resolution
spectra for a sample of 9 ACEPs belonging to the Galactic halo. We derived the
abundances of 12 elements, including C, Na, Mg, Si, Ca, Sc, Ti, Cr, Fe, Ni, Y,
and Ba. We complemented these data with literature abundances for an additional
three ACEPs that were previously incorrectly classified as type II Cepheids,
thus increasing the sample to a total of 12 stars. All the investigated ACEPs
have an iron abundance [Fe/H]$<-1.5$ dex as expected from theoretical
predictions for these pulsators. The abundance ratios of the different elements
to iron show that the ACEP's chemical composition is generally consistent with
that of the Galactic halo field stars, except the Sodium, which is found
overabundant in 9 out of the 11 ACEPs where it was measured, in close
similarity with second-generation stars in the Galactic Globular Clusters. The
same comparison with dwarf and ultra-faint satellites of the Milky Way reveals
more differences than similarities so it is unlikely that the bulk of Galactic
ACEPs originated in such a kind of galaxies which subsequently dissolved in the
Galactic halo. The principal finding of this work is the unexpected
overabundance of Sodium in ACEPs. We explored several hypotheses to explain
this feature, finding that the most promising scenario is the evolution of
low-mass stars in a binary system with either mass transfer or merging.
Detailed modelling is needed to confirm this hypothesis.
