A classification of $\mathbb{R}$-Fuchsian subgroups of Picard modular groups

Given an imaginary quadratic extension $K$ of $\mathbb{Q}$, we classify the maximal nonelementary subgroups of the Picard modular group $\mathrm{PU}(1,2;{\mathcal{O}}_K)$ preserving a totally real totally geodesic plane in the complex hyperbolic plane ${\mathbb{H}}_{\mathbb{C}}^2$. We prove that these maximal $\mathbb{R}$-Fuchsian subgroups are arithmetic, and describe the quaternion algebras from which they arise. For instance, if the radius $\mathrm{\Delta }$ of the corresponding $\mathbb{R}$-circle lies in $\mathbb{N}-\{0\}$, then the stabilizer arises from the quaternion algebra $\left(\text{\hspace{0.17em}⁣}\begin{array}{c}\mathrm{\Delta },\mathrm{\mid }{D}_K\mathrm{\mid }\\ \mathbb{Q}\end{array}\text{\hspace{0.17em}⁣}\right)$. We thus prove the existence of infinitely many orbits of $K$-arithmetic $\mathbb{R}$-circles in the hypersphere of ${\mathbb{P}}_2(\mathbb{C})$.