  We have found new anisotropic vacuum solutions for the scale-invariant
gravity theories which generalise Einstein's general relativity to a theory
derived from the Lagrangian $R^{1+\delta}$. These solutions are expanding
universes of Kasner form with an initial spacetime singularity at $t=0 $ and
exist for $-1/2<\delta <1/4$ but possess different Kasner index relations to
the classic Kasner solution of general relativity to which they reduce when
$\delta =0$. These solutions are unperturbed by the introduction of
non-comoving perfect-fluid matter motions if $p<\rho $ on approach to the
singularity and should not exhibit an infinite sequence of chaotic Mixmaster
oscillations when $\delta >0$.
