Emergence of the Cotton tensor for describing gravity

It is shown that the Cotton tensor can describe effects of gravity beyond general relativity. Any solution Einstein equations with or without cosmological constant satisfies field described by tensor. implies an integration constant. A vacuum a theory represented vanishing tensor, rather than Ricci An exact Schwarzschild-like for static and spherically symmetric source discovered. Although involve third order derivative, it found they reduce to second-order differential due variational principle.