New classes of groups which are equational domains

A group is CSA, if all of its maximal abelian subgroups are malnormal. It is known that every non-abelian CSA group is an equational domain. We generalize this result in two directions: we show that for a non-nilpotent group $G$ and a fixed positive integer $k$, if all maximal elements in the set of class $k$ nilpotent subgroups of $G$ are malnormal, then $G$ is an equational domain. Also, we prove that if a group $G$ is not locally nilpotent and if every maximal locally nilpotent subgroup of $G$ is malnormal, then $G$ is an equational domain.