  The t-J_z model is the strongly anisotropic limit of the t-J model which
captures some general properties of the doped antiferromagnets (AF). The
absence of spin fluctuations simplifies the analytical treatment of hole motion
in an AF background and allows us to calculate the single- and two-hole spectra
with high accuracy using regular diagram technique combined with real-space
approach. At the same time, numerical studies of this model via exact
diagonalization (ED) on small clusters show negligible finite size effects for
a number of quantities, thus allowing a direct comparison between analytical
and numerical results. Both approaches demonstrate that the holes have tendency
to pair in the p- and d-wave channels at realistic values of t/J. The
interactions leading to pairing and effects selecting p and d waves are
thoroughly investigated. The role of transverse spin fluctuations is considered
using perturbation theory. Based on the results of the present study, we
discuss the pairing problem in the realistic t-J-like model. Possible
implications for preformed pairs formation and phase separation are drawn.
