  Necessary conditions that the spurious state associated with the
translational motion and its double-phonon state have zero excitation energy in
extended RPA (ERPA) theories which include both one-body and two-body
amplitudes are investigated using the small amplitude limit of the
time-dependent density-matrix theory (STDDM). STDDM provides us with a quite
general form of ERPA as compared with other similar theories in the sense that
all components of one-body and two-body amplitudes are taken into account. Two
conditions are found necessary to guarantee the above property of the single
and double spurious states: The first is that no truncation in the
single-particle space should be made. This condition is necessary for the
closure relation to be used and is common for the single and double spurious
states. The second depends on the mode. For the single spurious state all
components of the one-body amplitudes must be included, and for the double
spurious state all components of one-body and two-body amplitudes have to be
included. It is also shown that the Kohn theorem and the continuity equations
for transition densities and currents hold under the same conditions as the
spurious states. ERPA theories formulated using the Hartree-Fock ground state
have a non-hermiticity problem. A method for formulating ERPA with hermiticity
is also proposed using the time-dependent density-matrix formalism.
