The purpose of this note is to prove the following result that left in the previous entry
Lemma. Provided the Weyl tensor vanishes, equation
is locally solvable if and only if the following integrability condition is satised
That is, if and only if the Cotton tensor vanishes.
Proof. It is necessary and suffcient to find a 1-form locally such that
where is a symmetric 2-tensor depending only on
and
. To see this, by the symmetry of the RHS, we have