A set $\Sigma$ of group identities is called bounded if there is $n\geq 1$ such that for any $(w\approx 1)\in \Sigma$, we have $w\in F(x\_1, \ldots, x\_n)$. A variety $\mathbf{V}$ is called bounded defined if $\mathbf{V}=Mod(\Sigma)$ for some bounded set $\Sigma$. 


Question: Is there an example of a bounded defined non-finitely based variety of groups?


P.S. $F(x\_1, \ldots, x\_n)$ is the free group of rank $n$.



Non finitely based varieties of groups defined by finitely many variables
Yes. It is usually called a *non-finitely based variety of finite axiomatic rank*.


Actually, one of the first examples of non-finitely based variety of groups has axiomatic rank two; [Adian proved](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=2442&option_lang=eng) that the following set of identities is independent:
$$
\{[x^{pn},y^{pn}]^n=1\;|\; p \hbox{ is prime}\},
$$
where $n\ge 1003$ is any given odd number. *Independence* means that none of the laws is a consequence of the other ones.


