  The Heisenberg double $D_q(E_2)$ of the quantum Euclidean group
$\mathcal{O}_q(E_2)$ is the smash product of $\mathcal{O}_q(E_2)$ with its Hopf
dual $U_q(\mathfrak{e}_2)$. For the algebra $D_q(E_2)$, explicit descriptions
of its prime, primitive, and maximal spectra are obtained. All prime factors of
$D_q(E_2)$ are presented as generalized Weyl algebras. As a result, we obtain
that the algebra $D_q(E_2)$ has no finite-dimensional representations, and that
$D_q(E_2)$ cannot have a Hopf algebra structure. The automorphism groups of the
quantum Euclidean group and its Heisenberg double are determined. Some
centralizers are explicitly described via generators and defining relations.
This enables us to give a classification of simple weight modules, and the
so-called $a$-weight modules, over the algebra $D_q(E_2)$.
