Independent axiomatizability of quasivarieties of torsion-free nilpotent groups

Let $N$ be a quasivariety of torsion-free nilpotent groups of class at most two. It is proved that the set of subquasivarieties in $N$, which have no independent basis of quasi-identities and are generated by a finitely generated group, is infinite. It is stated that there exists an infinite set of quasivarieties $M$ in $N$ which are generated by a finitely generated group and are such that for every quasivariety $K$ ($M\varsubsetneq K\subseteq N$), an interval $[M,K]$ has the power of the continuum in the quasivariety lattice.