Summary: For any integer \(n\neq 0,1\), a group is said to be \(n\)-Bell if it satisfies the law \([x^n,y]=[x,y^n]\). In this paper we prove that every finitely generated locally graded \(n\)-Bell group embeds into the direct product of a finite \(n\)-Bell group and a torsion-free nilpotent group of class \(\leq 2\). We prove that \(n\)-Bell groups which are not locally graded always have infinite simple sections of finite exponent. Additionally, we obtain similar results for the varieties of \(n\)-Levi groups and \(n\)-Abelian groups defined by the laws \([x^n,y]=[x,y]^n\) and \((xy)^n=x^ny^n\), respectively. We give characterizations of these groups in the locally graded case.