  Given an imaginary quadratic extension $K$ of $\mathbb Q$, we classify the
maximal nonelementary subgroups of the Picard modular group
$\operatorname{PU}(1,2;\mathcal O_K)$ preserving a totally real totally
geodesic plane in the complex hyperbolic plane $\mathbb H^2_\mathbb C$. 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 $\Delta$ of the corresponding $\mathbb R$-circle lies in $\mathbb
N-\{0\}$, then the stabilizer arises from the quaternion algebra
$\Big(\!\begin{array}{c} \Delta\,,\, |D_K|\\\hline\mathbb Q\end{array}
\!\Big)$. We thus prove the existence of infinitely many orbits of
$K$-arithmetic $\mathbb R$-circles in the hypersphere of $\mathbb P_2(\mathbb
C)$.
