Groups generated by (near) mutually engel periodic pairs

We use notations: [x,₁ y] = [x, y] and [x,_k+1 y] = [[x,_k y], y] for k ≥ 1. We consider groups generated by x,y satisfying relations x = [x,_n y],y = [y,_n x] or [x, y] = [x,_n y], [y, x] = [y,_n x]. We call groups of the first type mep-groups and of the second type nmep-groups. We show many properties and examples of mep- and nmep-groups. We prove that if p is a prime then the group Sl₂ (p) is a nmep-group. We give the necessary and sufficient conditions for metacyclic group to be a nmep-group and we show that nmep-groups with presentation {〈x,y|[x, y] = [x,₂ y],[y, x] = [y, ₂ x], xⁿ, y^m〉 are finite.