The conjugacy problem and virtually cyclic subgroups in the Artin braid group quotient $B_n\mathrm{/}[P_n,P_n]$

Let $n\ge 3$. In this paper we deal with the conjugacy problem in the Artin braid group quotient $B_n\mathrm{/}[P_n,P_n]$. To solve it we use systems of equations over the integers arising from the action of $B_n\mathrm{/}[P_n,P_n]$ over the abelianization of the pure Artin braid group $P_n\mathrm{/}[P_n,P_n]$. Using this technique we also realize explicitly infinite virtually cyclic subgroups in $B_n\mathrm{/}[P_n,P_n]$.