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[en] The equivariant version of the Curtis-Schori-West theorem is investigated. It is proved that for a nondegenerate Peano -continuum with an action of the compact abelian Lie group , the exponent is equimorphic to the maximal equivariant Hilbert cube if and only if the free part is dense in . We also show that the latter is sufficient for the equimorphy of and in the case of an action of an arbitrary compact Lie group . The key to the proof of these results lies in the theory of the universal -space (in the sense of Palais). Bibliography: 28 titles.