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This paper deals with the spatial structure of zero azimuthal wave number ULF oscillations in a 1-D inhomogeneous multi-component plasma when a finite ion gyrofrequency is taken into account. Such oscillations may occur in the terrestrial magnetosphere as Pc1-3 waves or in the magnetosphere of the planet Mercury. The wave field was found to have a sharp peak on some magnetic surfaces, an analogy of the Alfvén (field line) resonance in one-fluid MHD theory. The resonance can only take place for waves with frequencies in the intervals ω<ω<sub><i>ch</i></sub> or Ω<sub>0</sub><ω< ω<sub><i>cp</i></sub>, where ω<sub><i>ch</i></sub> and ω<sub><i>cp</i></sub> are heavy and light ions gyrofrequencies, and Ω<sub>0</sub> is a kind of hybrid frequency. Contrary to ordinary Alfvén resonance, the wave resonance under consideration takes place even at the zero azimuthal wave number. The radial component of the wave electric field has a pole-type singularity, while the azimuthal component is finite but has a branching point singularity on the resonance surface. The later singularity can disappear at some frequencies. In the region adjacent to the resonant surface the mode is standing across the magnetic shells.