  We demonstrate that in a single molecule magnet (SMM) strongly coupled to
electrodes the Kondo effect involves all magnetic excitations. This Kondo
effect is induced by the quantum tunneling of the magnetic moment (QTM).
Importantly, the Kondo temperature $T_K$ can be much larger than the magnetic
splittings. We find a strong modulation of the Kondo effect as function of the
transverse anisotropy parameter or a longitudinal magnetic field. For both
integer and half-integer spin this can be used for an accurate transport
spectroscopy of the magnetic states in low magnetic fields on the order of the
easy-axis anisotropy parameter. We set up a relationship between the Kondo
effects for successive integer and half-integer spins.
