  In this paper the solutions of the zero-sound dispersion equation in the
random phase approximation (RPA) are considered. The calculation of the damped
zero-sound modes \omega_s(k) (complex frequency of excitation) in the nuclear
matter is presented. The method is based on the analytical structure of the
polarization operators \Pi(\omega,k). The solutions of two dispersion equations
with \Pi(\omega,k) and with Re(\Pi(\omega,k)) are compared. It is shown that in
the first case we obtain one-valued smooth solutions without "thumb-like"
forms. Considering the giant resonances in the nuclei as zero-sound excitations
we compare the experimental energy and escape width of the giant dipole
resonance (GDR) in the nucleus A with \omega_s(k) taken at a definite wave
vector k=k_A.
