(Lobachevskii Journal of Mathematics, Vol.11, pp.27-38)

Let $G$ be a free product of two groups with amalgamated subgroup, $\pi$ be either the set of all prime numbers or the one-element set \{$p$\} for some prime number $p$. Denote by $\Sigma$ the family of all cyclic subgroups of group $G$, which are separable in the class of all finite $\pi$@-groups.\linebreak Obviously, cyclic subgroups of the free factors, which aren't separable in these factors by the family of all normal subgroups of finite $\pi$@-index of group $G$, the subgroups conjugated with them and all subgroups, which aren't $\pi ^{\prime}$@-isolated, don't belong to $\Sigma$. Some sufficient conditions are obtained for $\Sigma$ to coincide with the family of all other $\pi ^{\prime}$@-isolated cyclic subgroups of group $G$. /BR It is proved, in particular, that the residual $p$@-finiteness of a free product with cyclic amalgamation implies the $p$@-separability of all $p^{\prime}$@-isolated cyclic subgroups if the free factors are free or finitely generated residually $p$@-finite nilpotent groups