**AUTHORS:**Evgeny Astashov

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Equivariant maps, i.e., maps that commute with group actions on the source and target, play an important role in the study of manifolds with group actions. It is therefore of interest to classify equivariant maps up to certain equivalence relations. In this paper we study multivariate holomorphic function germs that are equivariant with respect to finite cyclic groups. The natural equivalence relation between such germs is provided by the action of the group of biholomorphic automorphism germs of the source. An orbit of this action is called equivariant simple if its sufficiently small neighborhood intersects only a finite number of other orbits. We present a sufficient condition under which there exist no singular equivariant holomorphic function germs; it is also shown that this condition is not necessary. The condition is formulated in terms of admissible sets of weights; such sets are defined and classified for all finite cyclic group representations. As an application we describe scalar actions of finite cyclic groups for which there exist no equivariant simple singular function germs.

**KEYWORDS:**
Equivariant maps, finite group actions, singularity theory, classification of singularities, simple singularities.

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