  It has been revealed through numerical calculations that the Second Random
Phase Approximation (SRPA) with the Hartree-Fock solution as its reference
state results in 1) spurious states at genuinely finite energy, contrary to
common expectation, and 2) unstable solutions, which within the first-order
Random Phase Approximation correspond to real low-energy collective vibrations.
In the present work, these shortcomings of SRPA are shown to not contradict
Thouless' theorem about the energy-weighted sum rule, and their origin is
traced to the violation of the stability condition. A more general theorem is
proven. Formal arguments are elucidated through numerical examples.
Implications for the validity of SRPA are discussed.
