What is known about the moduli space of stable rank $3$ bundles on the projective plane $\mathbb{CP}^2$? Ideally, there is a concrete complex manifold which is a fine moduli space for such bundles for some choice of topological type.

For rank $2$, that is explained in Section 4.1 in Chapter 2 of Okonek, Schneider, Spindler: Vector Bundles on Complex Projective Spaces. For arbitrary rank and second Chern class $c_2 <7$, there is Hulek: On the Classification of Stable Rank-r Vector Bundles over the Projective Plane, which relates stable bundles to stable Kronecker modules, but there is still a lot missing for my dream result.