On idempotents of semigroup varieties of $m$-groups

An m-group is a pair $(G,\varphi),$ where $G$ is an $\ell$-group and $\varphi$ is a decreasing order two automorphism of $G$. An $m$-group can be regarded as an algebraic system of signature $m$ and it is obvious that the $m$-groups form a variety in this signature. The set $M$ of varieties of all $m$-groups is a semigroup with respect to natural defined operation of multiplication of varieties. In this article we will give full description of idempotents of $M$.