  The behavior of strongly correlated Fermi systems is investigated beyond the
onset of a phase transition where the single-particle spectrum $\xi({\bf p})$
becomes flat. The Landau-Migdal quasiparticle picture is shown to remain
applicable on the ordered side of this transition. Nevertheless,
low-temperature properties evaluated within this picture show profound changes
relative to results of Landau theory, as a direct consequence of the flattening
of $\xi({\bf p})$. Stability conditions for this class of systems are examined,
and the nature of antiferromagnetic quantum phase transitions is elucidated.
