  An exotic fractionalized Fermi-liquid FL$^*$ theory of metallic systems,
which combines resonant-valence-bond (RVB) state and the band of current
carriers, is treated. An application of this theory to spin-liquid,
antiferromagnetic and nearly antiferromagnetic systems is proposed with the use
of various bosonic and fermionic representations, a comparison with
perturbation theory in the $s-d(f)$ exchange model being performed. The
topological aspects including formation of the small Fermi surface are treated.
In the case of narrow bands (strong correlations), the ground state is
considered as a direct product of RVB and dopon or Weng's fermion states.
Examples of Kondo lattices, doped pyrochlores, and metallic $\beta$-Mn,
YMn$_2$, Y$_{1-x}$Sc$_x$Mn$_2$ systems are discussed, analogies with
copper-oxide systems being treated.
