On the $\zeta_3$-Pell equation

Let $K = \mathbb{Q}(\zeta_3)$, where $\zeta_3$ is a primitive root of unity. In this paper we study the distribution of integers $\alpha \in \mathcal{O}_K$ for which the norm equation $N_{K(\sqrt[3]{\alpha})/K}(\mathbf{x}) = \zeta_3$ is solvable for integers $\mathbf{x} \in \mathcal{O}_{K(\sqrt[3]{\alpha})}$. The analogous question for $\zeta_2 = -1$ is the well-known negative Pell equation. We also address the natural generalization of Stevenhagen's conjecture on the negative Pell equation in this setting.