  We study singly-generated wavelet systems on $\Bbb R^2$ that are naturally
associated with rank-one wavelet systems on the Heisenberg group $N$. We prove
a necessary condition on the generator in order that any such system be a
Parseval frame. Given a suitable subset $I$ of the dual of $N$, we give an
explicit construction for Parseval frame wavelets that are associated with $I$.
We say that $g\in L^2(I\times \Bbb R)$ is Gabor field over $I$ if, for a.e.
$\lambda \in I$, $|\lambda|^{1/2} g(\lambda,\cdot)$ is the Gabor generator of a
Parseval frame for $L^2(\Bbb R)$, and that $I$ is a Heisenberg wavelet set if
every Gabor field over $I$ is a Parseval frame (mother-)wavelet for $L^2(\Bbb
R^2)$. We then show that $I$ is a Heisenberg wavelet set if and only if $I$ is
both translation congruent with a subset of the unit interval and dilation
congruent with the Shannon set.
