Existentially closed subgroups of free nilpotent groups

Let $\mathcal N_c$ be a variety of all nilpotent groups of class at most $c$, and let $N_{r,c}$ be a free nilpotent group of finite rank $r$ and nilpotency class $c$. It is proved that a subgroup $N$ of $N_{r,c}$ for $c\ge3$ is existentially closed in $N_{r,c}$ iff $N$ is a free factor of the group $N_{r,c}$ with respect to the variety $\mathcal N_c$. Consequently, $N\simeq N_{m,c}$, $1\le m\le r$ and $m\ge c-1$.