Geodesic
Verbal subgroups of complete direct products of groups
Matematičeskie zametki, Tome 9 (1971) no. 6, pp. 687-692.

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It is proved that if $V(X)$ is a proper verbal subgroup of a free group $X$ of countable rank, then a verbal subgroup $V(H)$ of the complete direct product $H=\widetilde\Pi^\times X_i$ of a countable number of isomorphic copies $X_i$ of $X$ differs from the complete direct product $\widetilde\Pi^\times V(X_i)$. 
@article{MZM_1971_9_6_a8,
     author = {S. A. Ashmanov},
     title = {Verbal subgroups of complete direct products of groups},
     journal = {Matemati\v{c}eskie zametki},
     pages = {687--692},
     publisher = {mathdoc},
     volume = {9},
     number = {6},
     year = {1971},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1971_9_6_a8/}
}
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S. A. Ashmanov. Verbal subgroups of complete direct products of groups. Matematičeskie zametki, Tome 9 (1971) no. 6, pp. 687-692. http://geodesic.mathdoc.fr/item/MZM_1971_9_6_a8/