  We study the superconducting properties of population-imbalanced ultracold
Fermi mixtures in one-dimensional (1D) optical lattices that can be effectively
described by the spin-imbalanced attractive Hubbard model (AHM) in the presence
of a Zeeman magnetic field. We use the mean-field theory approach to obtain the
ground state phase diagrams including some unconventional superconducting
phases such as the Fulde--Ferrell--Larkin--Ovchinnikov (FFLO) phase, and the
$\eta$ phase (an extremal case of the FFLO phase), both for the case of a fixed
chemical potential and for a fixed number of particles. It allows to determine
optimal regimes for the FFLO phase as well as $\eta$-pairing stability. We also
investigate the evolution from the weak coupling (BCS-like limit) to the strong
coupling limit of tightly bound local pairs (BEC) with increasing attraction,
at $T=0$. Finally, the obtained results show that despite of the occurrence of
the Lifshitz transition induced by an external magnetic field, the
superconducting state can still exist in the system, at higher magnetic field
values.
