### Groups Generated by (near) Mutually Engel Periodic Pairs

We use notations: $[x,y]=[{x}_{,1}y]$ and $[{x}_{,k+1}y]$ e $[[{x}_{,k}y],y]$. 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 $S{l}_{2}(p)$ 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 $\u27e8x,y\mid [x,y]=[{x}_{,2}y],[y,x]=[{y}_{,2}x],{x}^{n},{y}^{m}\u27e9$ are finite.