  With partially restored isospin symmetry, we calculate the nuclear matrix
element for a special decay mode of $2\nu\beta\beta$ (two neutrino double beta
decay) -- the decay to the first $2^+$ excited states. With the realistic
CD-Bonn nuclear force, we analyze the dependence of the nuclear matrix elements
on the iso-vector and iso-scalar parts of proton-neutron particle-particle
interaction. The dependence on the different nuclear matrix element is observed
and the results are explained. We also give the phase space factors with
numerical electron wave functions and properly chosen excitation energies.
Finally we give our results for the half-lives of this decay mode for different
nuclei.
