There are three levels of our study. First, we consider cases of classical varieties of alternative (Alt), Jordan (Jord), and Malcev (Malc) algebras. The results obtained at this level, having the form of new unpublished ones, in fact, accumulate the experience of certain known published examples of metabelian (two-step solvable) superalgebras and known bases for subspaces of multilinear polynomials in the free algebras of Alt, Jord, and Malc.

At its second level, the study deals with the case of the variety of all Lie-admissible algebras together with its proper subvarieties of flexible algebras, antiflexible algebras, and the algebras with the identity of Jacobian type for the associator function. The Theorems obtained at this level are new unpublished results giving the explicit descriptions of bases for U-superbimodules with no restrictions on sets of generators for U.

At the third level, we apply the techniques developed throughout the work for a finding of complete bases for the free superalgebras in certain nearly associative varieties that are also nearly nilpotent.