  The equations of motion of four-dimensional conformal gravity, whose
Lagrangian is the square of the Weyl tensor, require that the Bach tensor
$E_{\mu\nu}= (\nabla^\rho\nabla^\sigma + \ft12
R^{\rho\sigma})C_{\mu\rho\nu\sigma}$ vanishes. Since $E_{\mu\nu}$ is zero for
any Einstein metric, and any conformal scaling of such a metric, it follows
that large classes of solutions in four-dimensional conformal gravity are
simply given by metrics that are conformal to Einstein metrics (including
Ricci-flat). In fact it becomes more intriguing to find solutions that are {\it
not} conformally Einstein. We obtain five new such vacua, which are homogeneous
and have asymptotic generalized Lifshitz anisotropic scaling symmetry. Four of
these solutions can be further generalized to metrics that are conformal to
classes of pp-waves, with a covariantly-constant null vector. We also obtain
large classes of generalized Lifshitz vacua in Einstein-Weyl gravity.
