  An affine hypersurface M is said to admit a pointwise symmetry, if there
exists a subgroup G of Aut(T_p M) for all p in M, which preserves (pointwise)
the affine metric h, the difference tensor K and the affine shape operator S.
Here, we consider 3-dimensional indefinite affine hyperspheres, i.e. S= H Id
(and thus S is trivially preserved). First we solve an algebraic problem. We
determine the non-trivial stabilizers G of a traceless cubic form on a
Lorentz-Minkowski space R^3_1 under the action of the isometry group SO(1,2)
and find a representative of each SO(1,2)/G-orbit. Since the affine cubic form
is defined by h and K, this gives us the possible symmetry groups G and for
each G a canonical form of K. Next, we classify hyperspheres admitting a
pointwise G-symmetry for all non-trivial stabilizers G (apart from Z_2).
Besides well-known hyperspheres (for Z_2 x Z_2 resp. R the hyperspheres have
constant sectional curvature and Pick invariant J<0 resp. J=0 we obtain rich
classes of new examples e.g. warped product structures of two-dimensional
affine spheres (resp. quadrics) and curves. Moreover, we find a way to
construct indefinite affine hyperspheres out of 2-dimensional quadrics or
positive definite affine spheres.
