  The ground and first excited state of two superconducting layers in
interaction are studied considering two different coupling terms, one
represented by the standard Josephson interaction, and one new, which is a
superexchange pairing force between bilayer pairs. It is shown that a
moderate-to strong Josephson interaction produces a low-lying collective state,
pictured as an out-of-phase oscillation of the BCS gauge angles of the two
layers. This antisymmetric angular oscillation might explain the 41 meV
resonance observed in the neutron scattering experiments. The bilayer pairs are
formed by electrons from different layers with an antiparallel orientation of
the spins, being related to the antiferromagnetic arrangement. The pair
operators within the layers together with the bilayer pairs generate by
commutation an so(5) algebra. It is shown that the transition between the
superconducting and antiferromagnetic phases can be explained assuming the
dependence on concentration of the bilayer pairing strength, with maximum at
half-filling.
