$p$-solvability of regular equations over unitriangular groups over prime finite fields

An equation over a group with one unknown is called regular if the exponent sum of the unknown is nonzero. In this paper we prove that some regular equations of exponent $r{p}^s$, where $r\in \mathbb{Z}$, $s\in \mathbb{N}$, $\gcd (r,p)=1$, over the group UT${}_n({\mathbb{F}}_p)$ ($n\ge 2$) are solvable in an overgroup isomorphic to UT${}_{(n-1)p^s+1}({\mathbb{F}}_p)$. Applying this for $n=3$ we prove that any regular equation of exponent $r{p}^s$ over the Heisenberg $p$-group UT${}_3({\mathbb{F}}_p)$ is solvable in an overgroup isomorphic to UT${}_{2p^s+1}({\mathbb{F}}_p)$. The proofs of these results are constructive and allow to obtain solutions of equations in explicit form.