  Let $N$ be a step two connected and simply connected non commutative
nilpotent Lie group which is square-integrable modulo the center. Let $Z$ be
the center of $N$. Assume that $N=P\rtimes M$ such that $P$, and $M$ are simply
connected, connected abelian Lie groups, $M$ acts non-trivially on $P$ by
automorphisms and $\dim P/Z=\dim M$. We study band-limited subspaces of
$L^2(N)$ which admit Parseval frames generated by discrete translates of a
single function. We also find characteristics of band-limited subspaces of
$L^2(N)$ which do not admit a single Parseval frame. We also provide some
conditions under which continuous wavelets transforms related to the left
regular representation admit discretization, by some discrete set
$\Gamma\subset N$. Finally, we show some explicit examples in the last section.
