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We provide a new construction method for r-harmonic submanifolds in spheres, i. e. proper isometric r-harmonic immersions into spheres. Furthermore, we give new examples of r-harmonic spheres in spheres, possibly of high codimension. We determine index and nullity of these isometric r-harmonic immersions when r=2. Finally, we provide the normal index of these maps when r=3.
Balado-Alves et al. (Wed,) studied this question.