  We classify certain cases when the wreath products of distinct pairs of
groups generate the same variety. This allows us to investigate the
subvarieties of some nilpotent-by-abelian product varieties ${\mathfrak
U}{\mathfrak V}$ 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 same
length of solubility, and the same exponent, but which still are distinct
subvarieties. Obtained classification strengthens our recent work on varieties
generated by wreath products.
