The 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.