  We present a detailed study of the low-energy excitations of two existing
finite-size realizations of the planar kagome Heisenberg antiferromagnet on the
sphere, the cuboctahedron and the icosidodecahedron. After highlighting a
number of special spectral features (such as the presence of low-lying singlets
below the first triplet and the existence of localized magnons) we focus on two
major issues. The first concerns the nature of the excitations above the
plateau phase at 1/3 of the saturation magnetization Ms. Our exact
diagonalizations for the s=1/2 icosidodecahedron reveal that the low-lying
plateau states are adiabatically connected to the degenerate collinear
``up-up-down'' ground states of the Ising point, at the same time being well
isolated from higher excitations. A complementary physical picture emerges from
the derivation of an effective quantum dimer model which reveals the central
role of the topology and the intrinsic spin s. We also give a prediction for
the low energy excitations and thermodynamic properties of the spin s=5/2
icosidodecahedron Mo72Fe30. In the second part we focus on the low-energy
spectra of the s>1/2 Heisenberg model in view of interpreting the broad
inelastic neutron scattering response reported for Mo72Fe30. To this end we
demonstrate the simultaneous presence of several broadened low-energy ``towers
of states'' or ``rotational bands'' which arise from the large discrete spatial
degeneracy of the classical ground states, a generic feature of highly
frustrated clusters. This semiclassical interpretation is further corroborated
by their striking symmetry pattern which is shown, by an independent group
theoretical analysis, to be a characteristic fingerprint of the classical
coplanar ground states.
