Polyharmonic helices

The main aim of this paper is to investigate the existence of Frenet helices which are polyharmonic of order $r$, shortly, $r$-harmonic. We shall obtain existence, non-existence and classification results. More specifically, we obtain a complete classification of proper $r$-harmonic helices into the $3$-dimensional solvable Lie group Sol${}_3$. Next, we investigate the existence of proper $r$-harmonic helices into Bianchi-Cartan-Vranceanu spaces and, in this context, we find new examples. Finally, we shall establish some non-existence results both for Frenet curves and Frenet helices of order $n\ge 4$ when the ambient space is the Euclidean sphere ${\text{s}}^m$.