  A new type of self-similarity is found in the problem of a plane-parallel,
ultra-relativistic blast wave, propagating in a powerlaw density profile of the
form $\rho \propto z^{-k}$. Self-similar solutions of the first kind can be
found for $k<7/4$ using dimensional considerations. For steeper density
gradients with $k>2$, second type solutions are obtained by eliminating a
singularity from the equations. However, for intermediate powerlaw indices
$7/4<k<2$ the flow does not obey any of the known types of self-similarity.
Instead, the solutions belong to a new class in which the self-similar dynamics
are dictated by the non self-similar part of the flow. We obtain an exact
solution to the ultra-relativistic fluid equations and find that the non
self-similar flow is described by relativistic expansion into vacuum, composed
of (1) an accelerating piston that contains most of the energy and (2) a
leading edge of fast material that coincides with the interiors of the
blastwave and terminates at the shock. The dynamics of the piston itself are
self-similar and universal, and do not depend on the external medium. The exact
solution of the non self-similar flow is used to solve for the shock in the new
class of solutions.
