  In this paper we prove strong unique continuation for the following
degenerate elliptic equation \begin{equation}\label{e0} \Delta_zu
+|z|^2\partial_t^2u = Vu,\quad (z,t) \in \mathbb{R}^N \times \mathbb{R}
\end{equation} where the potential $V$ satisfies either of the following growth
assumptions \begin{align} & |V(z,t)| \leq \frac{f(\rho(z,t))}{\rho(z,t)^2},\
\text{where $f$ satisfies the Dini integrability condition as in (1.3)} \\ &
\text{or when } \notag \\ & |V(z,t)| \leq
C\frac{\psi(z,t)^{\epsilon}}{\rho(z,t)^2},\ \text{for some $\epsilon>0$ with
$\psi$ as in (2.6) and $N$ even.} \notag \end{align} This extends some of the
previous results obtained in [G] for this subfamily of Baouendi-Grushin
operators. As corollaries, we obtain new unique continuation properties for
solutions $u$ to \[ \Delta_{\mathbb{H}} u = Vu \] with certain symmetries as
expressed in (1.6) where $\Delta_{\mathbb{H}}$ corresponds to the sub-Laplacian
on the Heisenberg group $\mathbb{H}^n$.
