Primitive elements and automorphisms of the free metabelian group of rank $3$

The purpose of this article is twofold. On the one hand, we give an overview of the known results on primitive elements and automorphisms of a very specific free metabelian group $M_3$ of rank $3$. On the other hand, we present new results, at the same time showing the work of non-standard research tools of studying of this group.