  The t-J model is analysed in the limit of strong anisotropy, where the
transverse components of electron spin are neglected. We propose a
slave-particle-type approach that is valid, in contradiction to many of the
standard approaches, in the low-doping regime and becomes exact for a
half-filled system. We describe an effective method that allows to numerically
study the system with the no-double-occupancy constraint rigorously taken into
account at each lattice site. Then, we use this approach to demonstrate the
destruction of the antiferromagnetic order by increasing the doping and
formation of Nagaoka polarons in the strong interaction regime.
