The $n$-ary adding machine and solvable groups

We describe under various conditions abelian subgroups of the automorphism group $mathrm{Aut}(T_{n})$ of the regular $n$-ary tree $T_{n}$, which are normalized by the $n$-ary adding machine $tau =(e, dots, e,tau )sigma _{tau }$ where $sigma _{tau }$ is the $n$-cycle $left( 0,1, dots, n-1right) $. As an application, for $n=p$ a prime number, and for $n=4$, we prove that every soluble subgroup of $mathrm{Aut}(T_{n})$, containing $tau $ is an extension of a torsion-free metabelian group by a finite group.