For the multivalued logic $P_k$, where $k=p^r$, $p$ is a prime number and $r\ge2$, we describe some families of closed classes which contain the class $\operatorname{Pol}_k$ of polynomials and are contained in the class $\mathfrak M_k$ of the functions preserving congruence modulo $d=p^j$, $j=1,\dots,r-1$. These classes are closely related to subsets of the special complete system of the class $\mathfrak M_k$. We reveal a significant difference between the cases where $p=2$ and $p=3$. This research was supported by the Russian Foundation for Basic Research, grant 94–01–01206.