  A microscopic theory of linear response based on the Vlasov equation is
extended to systems having spheroidal equilibrium shape. The solution of the
linearized Vlasov equation, which gives a semiclassical version of the random
phase approximation, is studied for electrons moving in a deformed equilibrium
mean field. The deformed field has been approximated by a cavity of spheroidal
shape, both prolate and oblate. Contrary to spherical systems, there is now a
coupling among excitations of different multipolarity induced by the
interaction among constituents. Explicit calculations are performed for the
dipole response of deformed clusters of different size. In all cases studied
here the photoabsorption strength for prolate clusters always displays a
typical double-peaked structure. For oblate clusters we find that the
high--frequency component of the plasmon doublet can get fragmented in the
medium size region ($N \sim 250$). This fragmentation is related to the
presence of two kinds of three-dimensional electron orbits in oblate cavities.
The possible scaling of our semiclassical equations with the valence electron
number and density is investigated.
