The axiomatic rank of a quasivariety containing a free solvable group

Let $\mathfrak P$ be the class of groups for which the commutator subgroup of each subgroup intersects the center of that subgroup in the identity. In this paper it is shown that if a quasivariety $\mathfrak M$ contains a non-Abelian group free in $\mathfrak A^2$ and if $\mathfrak M\subseteq\mathfrak P$, then $\mathfrak M$ cannot be given by a system of quasi-identities in a finite number of variables.
Bibliography: 9 titles.