  We study type II universal metrics of the Lorentzian signature. These metrics
simultaneously solve vacuum field equations of all theories of gravitation with
the Lagrangian being a polynomial curvature invariant constructed from the
metric, the Riemann tensor and its covariant derivatives of an arbitrary order.
  We provide examples of type II universal metrics for all composite number
dimensions. On the other hand, we have no examples for prime number dimensions
and we prove the non-existence of type II universal spacetimes in five
dimensions.
  We also present type II vacuum solutions of selected classes of gravitational
theories, such as Lovelock, quadratic and L(Riemann) gravities.
