Special Lagrangians in nearly Kähler ${\mathbb{C}\mathbb{P}}^3$

This article explores special Lagrangian submanifolds in ${\mathbb{C}\mathbb{P}}^3$, viewed as a nearly Kähler manifold, from two different perspectives. Intrinsically, using a moving frame set-up, and extrinsically, using $\mathrm{S}\mathrm{U}(2)$ moment-type maps. We describe new homogeneous examples, from both perspectives, and classify totally geodesic special Lagrangian submanifolds. We show that every special Lagrangian in ${\mathbb{C}\mathbb{P}}^3$, or the flag manifold ${\mathbb{F}}_{1,2}({\mathbb{C}}^3)$ admitting a symmetry of an $\mathrm{S}\mathrm{U}(2)$ subgroup of nearly Kähler automorphisms is automatically homogeneous.