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We classify certain cases when the wreath products of distinct pairs ofgroups generate the same variety. This allows us to investigate thesubvarieties of some nilpotent-by-abelian product varieties ${\mathfrakU}{\mathfrak V}$ with the help of wreath products of groups. In particular,using wreath products we find such subvarieties in nilpotent-by-abelian${\mathfrak U}{\mathfrak V}$, which have the same nilpotency class, the samelength of solubility, and the same exponent, but which still are distinctsubvarieties. Obtained classification strengthens our recent work on varietiesgenerated by wreath products.

Subvariety structures in certain product varieties of groups