  We study an example of higher-order field-theoretic model with an
eighth-degree polynomial potential -- the $\varphi^8$ model. We show that for
some certain ratios of constants of the potential, the problem of finding
kink-type solutions in $(1+1)$-dimensional space-time reduces to solving
algebraic equations. For two different ratios of the constants, which determine
positions of the vacua, we obtained explicit formulas for kinks in all
topological sectors. The properties of the obtained kinks are also studied --
their masses are calculated, and the excitation spectra which could be
responsible for the appearance of resonance phenomena in kink-antikink
scattering are found.
