  We present an equation of state for elastic matter which allows for purely
longitudinal elastic waves in all propagation directions, not just principal
directions. The speed of these waves is equal to the speed of light whereas the
transversal type speeds are also very high, comparable to but always strictly
less than that of light. Clearly such an equation of state does not give a
reasonable matter description for the crust of a neutron star, but it does
provide a nice causal toy model for an extremely rigid phase in a neutron star
core, should such a phase exist. Another reason for focusing on this particular
equation of state is simply that it leads to a very simple recipe for finding
stationary rigid motion exact solutions to the Einstein equations. In fact, we
show that a very large class of stationary spacetimes with constant Ricci
scalar can be interpreted as rigid motion solutions with this matter source. We
use the recipe to derive a static spherically symmetric exact solution with
constant energy density, regular centre and finite radius, having a nontrivial
parameter that can be varied to yield a mass-radius curve from which stability
can be read off. It turns out that the solution is stable down to a tenuity R/M
slightly less than 3. The result of this static approach to stability is
confirmed by a numerical determination of the fundamental radial oscillation
mode frequency. We also present another solution with outwards decreasing
energy density. Unfortunately, this solution only has a trivial scaling
parameter and is found to be unstable.
