  We demonstrate that for some certain values of parameters of the
$(1+1)$-dimensional $\varphi^8$ model, the kink solutions can be found from
polynomial equations. For some selected values of the parameters we give the
explicit formulas for the kinks in all topological sectors of the model. Based
on the obtained algebraic equations, we show that in a special limiting case,
kinks with power-law asymptotics arise in the model, describing, in particular,
thick domain walls. Objects of this kind could be of interest for modern
cosmology.
