  This thesis explores nuclear systems under extreme conditions of isospin
asymmetry and strong magnetic fields. Its first chapter is devoted to the phase
diagram of isospin-asymmetrical nuclear matter in the density-temperature
plane. Four competing phases of matter are included into the consideration: the
unpaired phase, the ordinary Bardeen-Cooper-Schrieffer (BCS) phase, a phase
where normal fluid and superfluid separate into distinct domains and the
superfluid phase with a non-zero total momentum of Cooper pairs. The evolution
of these phases from the weakly BCS limit to the strongly coupled
Bose-Einstein-Condensate limit is studied by varying the density from high to
low values at various temperatures and asymmetries. The emergence of
tri-critical points (including a Lifshitz point) and various intrinsic
properties of the condensate (pair wave-function, occupation numbers,
quasiparticle spectra) are discussed. The second chapter studies the nuclei
$^{12}$C, $^{16}$O, and $^{20}$Ne in a magnetic field $B\approx 10^{17}\,$G by
utilizing the SKY3D code, suitably extended to include the magnetic field. Our
solutions of time-independent Hartree-Fock equations on a Cartesian
three-dimensional grid with a Skyrme interaction demonstrate a number of
interesting phenomena, such as level splitting in the magnetic field,
spontaneous rearrangement of energy levels, changes in the nuclear shape, etc.
The third chapter studies the pairing in low-density neutron matter in the
presence of a strong magnetic field. In particular, we find the limiting field
for the destruction of the superfluidity of neutrons in the crust and outer
core of magnetars. This field is density-dependent but is generally of the
order of $B\geq 10^{17}\,$G. The phase diagram of spin-polarized neutron matter
is constructed in the case where only the BCS phase is allowed along with the
(spin-polarized) normal phase.
