The `post-Bianchi equations' and the integrability of the vacuum Einstein equations

The `post-Bianchi equations', i.e., the extra integrability conditions that arise when a choice of frame is made, provide additional constraints on the mathematical problem of integrating the vacuum Einstein equations. It is shown that these equations give an expression for the divergence of the connection in terms of the Newman - Penrose variables. Explicit expressions for the spin-coefficients, satisfying the Bianchi identities, are also given. A matrix integrability theorem is formulated and used to transform the Einstein equations into a new matrix form.