  An exactly solvable model for a description of the two-neutrino double beta
decay transition of the Fermi type is considered. By using perturbation theory
an explicit dependence of the two-neutrino double beta decay matrix element on
the like-nucleon pairing, particle-particle and particle-hole proton-neutron
interactions by assuming a weak violation of isospin symmetry of Hamiltonian
expressed with generators of the SO(5) group. It is found that there is a
dominance of double beta decay transition through a single state of the
intermediate nucleus. Then, an energy weighted sum rule connecting Delta Z=2
nuclei is presented and discussed. It is suggested that this sum rule can be
exploited to study the residual interactions of the nuclear Hamiltonian.
