  Paired state of nonstandard quasiparticles is analyzed in detail in two model
situations. Namely, we consider the Cooper-pair bound state and the condensed
phase of an almost localized Fermi liquid (ALFL) composed of quasiparticles in
a narrow-band with the spin-dependent masses (SDM) and an effective field, both
introduced earlier and induced by strong electronic correlations. Each of these
novel characteristics are calculated in a self-consistent manner. We analyze
the bound states as a function of Cooper-pair momentum q in applied magnetic
field in the strongly Pauli limiting case (i.e. when the orbital effects of
applied magnetic field are disregarded). The spin-direction dependence of the
effective mass makes the quasiparticles comprising Cooper pair spin
distinguishable in the quantum mechanical sense, whereas the condensed gas of
pairs may still be regarded as composed of identical entities. The
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) condensed phase of moving pairs is by
far more robust in the applied field for the case with spin-dependent masses
than in the situation with equal masses of quasiparticles. Relative stability
of the Bardeen-Cooper-Schrieffer (BCS) vs. FFLO phase is analyzed in detail on
temperature - applied field plane. Although our calculations are carried out
for a model situation, we can conclude that the spin-dependent masses should
play an important role in stabilizing high-field low-temperature (HFLT)
unconventional superconducting phases (FFLO being an instance) in systems such
as CeCoIn_5, organic metals, and possibly others.
