Dipole—exchange spin excitations in composite ‘nanorice’-type nanoparticle (as oblate spheroid) are theoretically investigated. A nanorice with a nonmagnetic core and a shell composed of a uniaxial ferromagnet with the local ‘easy axis’ type is considered; dissipation effects are neglected. Spin dynamics in the above-mentioned nanosystem is described by the linearized Landau—Lifshitz equation (within the magnetostatic approximation) with the addends, which allow for the magnetic dipole—dipole interaction, the exchange interaction, and the anisotropy effects. Considering the system symmetry, a prolate spheroidal co-ordinate system is used. After using one of the Maxwell equations, magnetization in the Landau—Lifshitz equation is eliminated, and an equation for the magnetic potential of the above-mentioned spin excitations is obtained. A solution for the above-mentioned equation is proposed in the form of a combination of the generalized spheroidal functions; this combination cannot be considered as the solution of the equation in general case, however, it can be considered as an approximate solution for the case of a thin shell with the internal ellipsoid semi-axes ratio close to one. For the above-described case, a dispersion relation for such spin excitations is also found.