  We report on a spectroscopic study of the spin-wave eigen-modes inside an
individual normally magnetized two layers circular nano-pillar
(Permalloy$|$Copper$|$Permalloy) by means of a Magnetic Resonance Force
Microscope (MRFM). We demonstrate that the observed spin-wave spectrum
critically depends on the method of excitation. While the spatially uniform
radio-frequency (RF) magnetic field excites only the axially symmetric modes
having azimuthal index $\ell=0$, the RF current flowing through the
nano-pillar, creating a circular RF Oersted field, excites only the modes
having azimuthal index $\ell=+1$. Breaking the axial symmetry of the
nano-pillar, either by tilting the bias magnetic field or by making the pillar
shape elliptical, mixes different $\ell$-index symmetries, which can be excited
simultaneously by the RF current. Experimental spectra are compared to
theoretical prediction using both analytical and numerical calculations. An
analysis of the influence of the static and dynamic dipolar coupling between
the nano-pillar magnetic layers on the mode spectrum is performed.
