  The two-neutrino double-$\beta$ Gamow-Teller and Fermi transitions are
studied within an exactly solvable model, which allows a violation of both
spin-isospin SU(4) and isospin SU(2) symmetries, and is expressed with
generators of the SO(8) group. It is found that this model reproduces the main
features of realistic calculation within the quasiparticle random-phase
approximation with isospin symmetry restoration concerning the dependence of
the two-neutrino double-$\beta$ decay matrix elements on isovector and
isoscalar particle-particle interactions. By using perturbation theory an
explicit dependence of the two-neutrino double-$\beta$ decay matrix elements on
the like-nucleon pairing, particle-particle T=0 and T=1, and particle-hole
proton-neutron interactions is obtained. It is found that double-$\beta$ decay
matrix elements do not depend on the mean field part of Hamiltonian and that
they are governed by a weak violation of both SU(2) and SU(4) symmetries by the
particle-particle interaction of Hamiltonian. It is pointed out that there is a
dominance of two-neutrino double-$\beta$ decay transition through a single
state of intermediate nucleus. The energy position of this state relative to
energies of initial and final ground states is given by a combination of
strengths of residual interactions. Further, energy-weighted Fermi and
Gamow-Teller sum rules connecting Delta Z = 2 nuclei are discussed. It is
proposed that these sum rules can be used to study the residual interactions of
the nuclear Hamiltonian, which are relevant for charge-changing nuclear
transitions.
