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