In the context of the white dwarf asteroseismology, we investigatevibrational properties of a non-convective solid star with an axisymmetricpurely toroidal intrinsic magnetic field of two different shapes. Focus is laidon regime of node-free global Lorentz-force-driven vibrations about symmetryaxis at which material displacements have one and the same form as those fornodeless spheroidal and torsional vibrations restored by Hooke's force ofelastic shear stresses. Particular attention is given to the even-paritypoloidal Alfven modes whose frequency spectra are computed in analytic formshowing how the purely toroidal magnetic fields completely buried beneath thestar surface can manifest itself in seismic vibrations of non-magnetic whitedwarfs. The obtained spectral formulae are discussed in juxtaposition withthose for Alfven modes in the solid star model with the poloidal, homogeneousinternal and dipolar external, magnetic field whose inferences are relevant toAlfven vibrations in magnetic white dwarfs.
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