  Structural and thermodynamic properties of single-crystalline
UNi$_{1-x}$Ge$_2$ with $x$\,=\,0.66 have been investigated by measuring
magnetization, specific heat, and thermal expansion over a wide range of
temperatures and magnetic fields. The measurements revealed the emergence of a
long-range antiferromagnetic ordering of uranium magnetic moments below the
N{\'e}el temperature $T_{\rm N}$\,=\,45.5\,K and the existence of two easy axes
in the studied compound, namely $b$ and $c$, which correspond to the planes of
the uranium zig-zag chains. Magnetic field applied along these two
crystallographic directions induces in the system a first-order metamagnetic
phase transition (from antiferromagnetism to field-polarized paramagnetism),
and the width of the magnetic hysteresis associated with that transition
reaches as much as 40 kOe at the lowest temperatures. A magnetic phase diagram
developed from the experimental data showed that the metastable region
associated with that magnetic hysteresis forms a funnel that narrows toward the
N{\'e}el point in zero magnetic field. The four-layer Ising model has
successfully predicted the collinear antiferromagnetic structure in
UNi$_{0.34}$Ge$_2$ (known from earlier reports), its magnetic phase diagram,
and temperature and field variations of its magnetization. Moreover, it
suggests that the first-order phase transition extends down to zero magnetic
field, although it is barely detectable in the experiments performed in low
magnetic fields. According to this model, the second-order phase transition
occurs in the compound only in zero field.
