  A perturbative renormalization group analysis of interacting electrons on a
two-leg ladder reveals that at half-filling any weakly repulsive system scales
onto an exactly soluble Gross-Neveu model with a hidden SO(8) symmetry. The
half-filled ground state is a Mott insulator with short-range d-wave pair
correlations. We extract the exact energies, degeneracies, and quantum numbers
of *all* the low energy excited multiplets. One energy (mass) m octets contains
Cooper pair, magnon, and density-wave excitations, two more octets contain
single-particle excitations, and a mass \sqrt{3}m antisymmetric tensor contains
28 "bound states". Exact single-particle and spin gaps are found for the
lightly-doped (d-wave paired one-dimension Bose fluid) system. We also
determine the four other robust phases occuring at half-filling for partially
attractive interactions. All 5 phases have distinct SO(8) symmetries, but share
S.C. Zhang's SO(5) as a common subgroup.
