  Compared to isolated C$_{60}^{3-}$ ions, characterized by a threedimensional
equipotential trough at the bottom of the lowest adiabatic potential energy
surface (APES), the Jahn-Teller (JT) effect in cubic fullerides is additionally
influenced by the interaction of JT distortions at C$_{60}$ sites with
vibrational modes of the lattice. This leads to modification of JT
stabilization energy and to the warping of the trough at each fullerene site,
as well as to the interaction of JT distortions at different sites. Here we
investigate these effects in three fcc fullerides with A=K,Rb,Cs and in
Cs$_3$C$_{60}$ with bcc (A15) structure. DFT calculations of orbital vibronic
coupling constants at C$_{60}$ sites and of phonon spectra have been done for
fully ordered lattices (1 C$_{60}$/u.c.). Based on them the elastic response
function for local JT distortions has been evaluated and the lowest APES
investigated. To this end an expression for the latter in function of trough
coordinates of all sites has been derived. The results show that the JT
stabilization energy slightly increases compared to an isolated C$_{60}^{3-}$
and a warping of the trough of few meV occurs. The interaction of JT
distortions on nearest- and next-nearest-neighbor fullerene sites is of similar
order of magnitude. These effects arise first of all due to the interaction of
C$_{60}$ sites with the displacements of neighbor alkali atoms and are more
pronounced in fcc fullerides than in the A15 compound. The results of this
study support the picture of weakly hindered independent rotations of JT
deformations at C$_{60}$ sites in cubic A$_3$C$_{60}$.
