Generalized rigid groups: definitions, basic properties, and problems

We find a natural generalization of the concept of rigid group. The generalized rigid groups are also called $r$-groups. The terms of the corresponding rigid series of every $r$-group can be characterized by both $\exists$-formulas and $\forall$-formulas. We find a recursive system of axioms for the class of $r$-groups of fixed solubility length. We define divisible $r$-groups and give an appropriate system of axioms. Several fundamental problems are stated.