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.
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