$G$-Identities of Nilpotent Groups. I

The structure of a group $V_{n,red}(G)$ of reduced $G$-identities of rank $n$ is treated subject to the condition that $G$ is a nilpotent group of class 1 or 2. The results obtained allow us to settle the question of whether a $G$-variety $G$-$\operatorname{var}(G)$ generated by a nilpotent group $G$ of class 2 is finitely based. Moreover, we introduce the concepts of a $d$-commutator subgroup and of a main $d$-group, associated with $G$.