Regular rigid Korovin orbits

An example of an infinite regular feebly compact quasitopological group is presented such that all continuous real-valued functions on the group are constant. The example is based on the use of Korovin orbits in $X^G$, where $X$ is a special regular countably compact space constructed by S.Bardyla and L.Zdomskyy and $G$ is an abstract Abelian group of an appropriate cardinality. Also, we study the interplay between the separation properties of the space $X$ and Korovin orbits in $X^G$. We show in particular that if $X$ contains two nonempty disjoint open subsets, then every Korovin orbit in $X^G$ is Hausdorff.