Description of coordinate groups of irreducible algebraic sets over free 2-nilpotent groups

A convenient pure algebraic description of the coordinate groups of irreducible algebraic sets over a non-Abelian free 2-nilpotent group $N$ of finite rank is given. Note that, in algebraic geometry over an arbitrary group $N$, it is natural to consider groups containing $N$ as a subgroup (so-called $N$-groups) and homomorphisms of $N$-groups which are identical on $N$ ($N$-homomorphisms). As a corollary, we describe all finitely generated groups $H$ that are universally equivalent to $N$ (with constants from $N$ in the language). Additionally, we give a pure algebraic criterion determining when a finitely generated $N$-group $H$ that is $N$-separated by $N$ is, in fact, $N$-discriminated by $N$.