Definability in algebraically closed groups

Let K be an abstract class of groups such that a countable group U exists possessing the following properties: 1) an arbitrary finitely generated subgroup of U belongs to K; 2) an arbitrary finitely generated subgroup from K is imbedded in U; 3) a recursive representaion of the group U exists with a solvable word identity problem. Then for arbitrary n ≥ 1 there exists ??-equation Ψ_n(v₀...v_n?1) such that for an arbitrary algebraically closed group G and for arbitrary x₀...x_n?1 ε G Classes of finite free nilpotent groups satisfy the conditions of the theorem.