The purpose of this note is to prove the following result that left in the previous entry

Lemma. Provided the Weyl tensor vanishes, equationis 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