Absolutely Closed Groups in the Class of $2$-Step Nilpotent Torsion-Free Groups

It is proved that divisible groups and only these groups are absolutely closed (with respect to the operator of dominion) in the class of $2$-step nilpotent torsion-free groups. It is established that the additive group of the rationals is $1$-closed in an arbitrary quasivariety of nilpotent torsion-free groups and $3$-closed in an arbitrary quasivariety of $2$-step nilpotent torsion-free groups.