On varieties with identities of one generated free metabelian algebra

A set of linear algebras where a fixed set of identities takes place, following A.I. Maltsev, is called a variety. In the case of zero characteristic of the main field all the information about the variety is contained in multilinear parts of relatively free algebra of the variety. We can study the identities of variety by means of investigations of multilinear part of degree $n$ as module of the symmetric group $S_n.$
Using the language of Lie algebras we say that an algebra is metabelian if it satisfies the identity $(xy)(zt)\equiv 0.$
In this paper we study the identities of non-associative one-generated free metabelian algebra and its factors. In particular, the infinite set of the varieties with different fractional exponents between one and two was constructed.
Note that the sequence of codimensions of these varieties asymptotically formed by using colength, and not by using the dimension of some irreducible module of the symmetric group what was for all known before examples.
Bibliography: 18 titles.