On the Conjugacy Problem in the Group $F/N_1\cap N_2$

Let $N_1$ ($N_2$) be the normal closure of a finite symmetrized set $R_1$ ($R_2$, respectively) in a finitely generated free group $F=F(A)$. As is known, if $R_i$ satisfies condition $C(6)$, then the conjugacy problem is decidable in $F/N_i$. In the paper, it is proved that, if one adds to condition $C(6)$ on the set $R_1\cup R_2$ the atoricity condition for the presentation $\langle A\mid R_1,R_2\rangle$, then the conjugacy problem is decidable in the group $F/N_1\cap N_2$ as well. In particular, for the decidability of the conjugacy problem in $F/N_1\cap N_2$, it is sufficient to assume that the set $R_1\cup R_2$ satisfies condition $C(7)$.