The primitive ideal space of the partial-isometric crossed product corresponding to an action induced by the multiplication on $p$-adic integers

For an odd prime $p$, we consider an action $\alpha$ of the semigroup $\mathbb{N}^{2}$ on the algebra $C(\mathbb{Z}_p)$ induced by the multiplication on the compact topological ring of $p$-adic integers $\mathbb{Z}_p$.

We then identify all primitive ideals of the partial-isometric crossed product $C(\mathbb{Z}_{p})\rtimes_{\alpha}^{\textrm{piso}} \mathbb{N}^{2}$ and describe the hull-kernel closures of the subsets of its primitive ideal space.