  In the context of three-dimensional conformal higher spin theory we derive,
in the frame field formulation, the full non-linear spin 3 Cotton equation
coupled to spin 2. This is done by solving the corresponding Chern-Simons gauge
theory system of equations, that is, using $F=0$ to eliminate all auxiliary
fields and thus expressing the Cotton equation in terms of just the spin 3
frame field and spin 2 covariant derivatives and tensors (Schouten). In this
derivation we neglect the spin 4 and higher spin sectors and approximate the
star product commutator by a Poisson bracket. The resulting spin 3 Cotton
equation is complicated but can be related to linearized versions in the metric
formulation obtained previously by other authors. The expected symmetry (spin 3
"translation", "Lorentz" and "dilatation") properties are verified for Cotton
and other relevant tensors but some perhaps unexpected features emerge in the
process, in particular in relation to the non-linear equations. We discuss the
structure of this non-linear spin 3 Cotton equation but its explicit form is
only presented here, in an exact but not completely refined version, in
appended files obtained by computer algebra methods. Both the frame field and
metric formulations are provided.
