Exact solutions of Cotton Gravity in its Codazzi formulation

The "Codazzi formulation", based on a Codazzi tensor, provides a more robust and straightforward theoretical framework for "Cotton Gravity" (CG) than its original formulation in terms of the Cotton tensor. Using this formulation we provide a self-consistent procedure to generate non-trivial exact solutions in CG that generalize well known General Relativity (GR) solutions. We re-derive a known CG solution that generalizes the Schwarzschild solution of GR, showing that it is the unique vacuum solution of static spherical symmetry in CG, extending this result to a CG generalization of the Reissner-Nordstrom solution of GR, all of which places a strong case supporting the fulfillment of Birkhoff's theorem. When applied to Friedman-Lema\^\i tre-Robertson-Walker (FLRW) models CG naturally identifies the $\Lambda$CDM model as the unique FLRW dust model with constant negative spatial curvature. We also obtain CG generalizations of the Lema\^\i tre-Tolman-Bondi (LTB) and Szekeres dust solutions of GR, allowing for time and space dependent changes from decelerated to accelerated evolution, without necessarily assuming a dark energy source. The CG generalization of static perfect fluid spheres allows in the weak field regime to model the flattening of rotation velocities in spherical galactic systems without assuming dark matter. We also generalize non-static spherically symmetric perfect fluid solutions with a shear-free 4 velocity. Our results suggest the need for further research using the Codazzi formulation to explore the potential for applications of CG to current open problems in gravitational systems.