  In two-dimensional systems possessing a high degree of symmetry, the
repulsive electron-electron interaction produces a pairing force; the mechanism
would fail in the presence of strong distortions. We have studied this in the
one-band and three-band Hubbard Model. From partially occupied orbitals one
obtains pair eigenstates of the Hamiltonian with no on-site repulsion (the W=0
pairs). The concept of W=0 pairs allows to make qualitative and quantitative
predictions about the behaviour of interacting many-body systems, a quite
remarkable and unusual situation. Exact numerical solutions for clusters with
``magic'' hole numbers reveal attraction between the holes in W=0 pairs. The
effect occurs in all fully symmetric clusters which are centered on a Cu site;
then holes get paired in a wide, physically relevant parameter range and show
superconducting quantization of the magnetic flux. A canonical transformation
of the Hamiltonian, valid for clusters and for the full plane, leads to a
Cooper-like equation for the W=0 pairs. We have evaluated the effective
interaction and found that W=0 pairs are the bare quasiparticles which, once
dressed, become two-hole bound states. We applied the above theory to the doped
antiferromagnet, and found that the ground state at half filling is the singlet
component of a determinantal state. We write down this determinant and the
ground state wave function explicitly in terms of a many-body W=0 eigenstate.
Our analytical results for the $4\times 4$ square lattice at half filling and
with doped holes, compared to available numerical data, demonstrate that the
method, besides providing intuitive grasp on pairing, also has quantitative
predictive power.
