Abstract
Bose-Einstein condensation of ultra cold atoms is typically realized in magnetic traps which effectively lead to an axially symmetric harmonic potential. This letter shows that the spectrum of collective vibrational modes of a repulsive condensate in a prolate potential displays a defect known as quantum monodromy. The monodromy is analysed on the basis of the dynamics of quasiparticles. In terms of the quasiparticles the regime of collective modes or the so-called hydrodynamic regime is characterized through kinetic energies much smaller than the chemical potential. In this limit the classical dynamics of the quasiparticles is integrable. The monodromy is quantitatively described by a monodromy matrix that is calculated from classical actions.