Geodesic
On the definability of endomorphisms of a free group of the variety $\mathfrak{AM}$ by a finite set of values
Matematičeskie zametki, Tome 62 (1997) no. 6, pp. 916-920.

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In the paper we find out in what cases any endomorphism of a free metabelian group of rank $n$ is uniquely determined by its values on finitely many elements of the group. 
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E. I. Timoshenko. On the definability of endomorphisms of a free group of the variety $\mathfrak{AM}$ by a finite set of values. Matematičeskie zametki, Tome 62 (1997) no. 6, pp. 916-920. http://geodesic.mathdoc.fr/item/MZM_1997_62_6_a12/

[1] Kourovskaya tetrad. Nereshennye zadachi teorii grupp, Novosibirsk, 1995

[2] Neiman Kh., Mnogoobraziya grupp, Mir, M., 1969

[3] Lewin J. A., “A note on zero divisors in group-ring”, Proc. Amer. Math. Soc., 31:2 (1972), 357–359 | DOI | MR | Zbl