  We use vector Hamiltonian formalism (VHF) to study theoretically three-magnon
parametric interaction (or three-wave splitting) in a magnetic disk existing in
a magnetic vortex ground state. The three-wave splitting in a disk is found to
obey two selection rules: (i) conservation of the total azimuthal number of the
resultant spin-wave modes, and (ii) inequality for the radial numbers of
interacting modes, if the mode directly excited by the driving field is
radially symmetric (i.e. if the azimuthal number of the directly excited mode
is $m = 0$). The selection rule (ii), however, is relaxed in the "small"
magnetic disks, due to the influence of the vortex core. We also found, that
the efficiency of the three-wave interaction of the directly excited mode
strongly depends on the azimuthal and radial mode numbers of the resultant
modes, that becomes determinative in the case when several splitting channels
(several pairs of resultant modes) simultaneously approximately satisfy the
resonance condition for the splitting. The good agreement of the VHF analytic
calculations with the experiment and micromagnetic simulations proves the
capability of the VHF formalism to predict the actual splitting channels and
the magnitudes of the driving field thresholds for the three-wave splitting.
