A growing number of dynamical situations involve the coupling of particles or
singularities with physical waves. In principle these situations are very far
from the wave-particle duality at quantum scale where the wave is probabilistic
by nature. Yet some dual characteristics were observed in a system where a
macroscopic droplet is guided by a pilot-wave it generates. Here we investigate
the behaviour of these entities when confined in a two-dimensional harmonic
potential well. A discrete set of stable orbits is observed, in the shape of
successive generalized Cassinian-like curves (circles, ovals, lemniscates,
trefoils...). Along these specific trajectories, the droplet motion is
characterized by a double quantization of the orbit spatial extent and of the
angular momentum. We show that these trajectories are intertwined with the
dynamical build-up of central wave-field modes. These dual self-organized modes
form a basis of eigenstates on which more complex motions are naturally
decomposed
