We analyze the ground states and the elementary collective excitations (phonons) of a class of systems, which form cluster crystals in the absence of attractions. Whereas the regime of moderate-to-high temperatures in the phase diagram has been analyzed in detail by means of density functional considerations (Likos et al 2007 J. Chem. Phys. 126 224502), the present approach focuses on the complementary regime of low temperatures. We establish the existence of an infinite cascade of isostructural transitions between crystals with different lattice site occupancies at T = 0 and we quantitatively demonstrate that the thermodynamic instabilities are bracketed by mechanical instabilities arising from long-wavelength acoustical phonons. We further show that all optical modes are degenerate and flat, giving rise to almost perfect realizations of Einstein crystals. We calculate analytically the complete phonon spectrum for the whole class of models as well as the Helmholtz free energy of the systems. On the basis of the latter, we demonstrate that the aforementioned isostructural phase transitions must terminate at an infinity of critical points at low temperatures, brought about by the anharmonic contributions in the Hamiltonian and the hopping events in the crystals.