  We consider collective excitations of a Fermi liquid. For each value of the
angular momentum $l$, we study the evolution of longitudinal and transverse
collective modes in the charge (c) and spin (s) channels with the Landau
parameter $F_l^{c(s)}$, starting from positive $F_l^{c(s)}$ and all the way to
the Pomeranchuk transition at $F_l^{c(s)} = -1$. In each case, we identify a
critical zero-sound mode, whose velocity vanishes at the Pomeranchuk
instability. For $F_l^{c(s)} < -1$, this mode is located in the upper frequency
half-plane, which signals an instability of the ground state. In a clean Fermi
liquid the critical mode may be either purely relaxational or almost
propagating, depending on the parity of $l$ and on whether the response
function is longitudinal or transverse. These differences lead to qualitatively
different types of time evolution of the order parameter following an initial
perturbation. A special situation occurs for the $l = 1$ order parameter that
coincides with the spin or charge current. In this case the residue of the
critical mode vanishes at the Pomeranchuk transition. However, the critical
mode can be identified at any distance from the transition, and is still
located in the upper frequency half-plane for $F_1^{c(s)} < -1$. The only
peculiarity of the charge/spin current order parameter is that its time
evolution occurs on longer scales than for other order parameters. We also
analyze collective modes away from the critical point, and find that the modes
evolve with $F_l^{c(s)}$ on a multi-sheet Riemann surface. For certain
intervals of $F_l^{c(s)}$, the modes either move to an unphysical Riemann sheet
or stay on the physical sheet but away from the real frequency axis. In that
case, the modes do not give rise to peaks in the imaginary parts of the
corresponding susceptiblities.
