  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.
