[en] Equations, derived early to describe the collective-quadrupole-oscillation spectrum of fast rotating nuclei, are reduced to the canonical (Hamiltonian) form and quantized. Probabilities of the quadrupole transitions between the vibrational states, constructed on spheroidal equilibrium configurations, are studied. A strong spin dependence of the position and electromagnetic properties of such states is predicted. A possibility of the quadrupole transitions along a sequence of levels near the yrast line in nuclei, rotating around the symmetry axis, is shown