This paper is concerned with the transverse structure of the ion-ion hybrid (IIH) modes in dipole magnetospheric plasma enriched with heavy ions and inhomogeneous both across the magnetic shells and along the field lines. It was found that the waves can be locked in the transverse resonator between two cut-off surfaces. The wave's energy in the resonator is concentrated near the geomagnetic equator and across the magnetic shells and propagates in the azimuthal direction. The existence of the transverse resonator results from the difference of the frequencies of the field line oscillations in the radial and azimuthal directions (poloidal and toroidal eigenfrequencies, respectively) and is ultimately caused by the field line curvature. The resonator can be located on the plasmapause where the Alfvén speed has a local minimum, or near the local minimum of the heavy ion's density. The transverse width of the resonator is estimated to be approximately 0.3 RE, and the wave's frequency is about 0.3 Hz, which correspond to the parameters of Pc1 waves observed in the Earth's magnetosphere.