  Let $(A,\mathbb{N}^{2},\alpha)$ be a dynamical system consisting of a
$C^*$-algebra $A$ and an action $\alpha$ of $\mathbb{N}^{2}$ on $A$ by
automorphisms. Let $A\times_{\alpha}^{\textrm{piso}}\mathbb{N}^{2}$ be the
partial-isometric crossed product of the system. We apply the fact that it is a
full corner of a crossed product by the group $\mathbb{Z}^{2}$ in order to give
a complete description of its primitive ideal space.
