Cawley's counterexample to Dirac's conjecture as a curved spacetime

Cawley's counterexample Lagrangian to Dirac's conjecture on dynamical systems is modified to a line element in curved spacetime, and the energy-momentum tensor corresponding to such a spacetime is found. The spacetime obtained satisfies the Einstein field equations and describes a three-dimensional matter-filled universe. It is further shown that such a universe cannot be filled up with other sources, such as a perfect fluid, a scalar field, or an electromagnetic field, without violating the Einstein field equations.