First Integrals of Extended Hamiltonians in $n+1$ Dimensions Generated by Powers of an Operator

We describe a procedure to construct polynomial in the momenta first integrals of arbitrarily high degree for natural Hamiltonians $H$ obtained as one-dimensional extensions of natural (geodesic) $n$-dimensional Hamiltonians $L$. The Liouville integrability of $L$ implies the (minimal) superintegrability of $H$. We prove that, as a consequence of natural integrability conditions, it is necessary for the construction that the curvature of the metric tensor associated with $L$ is constant. As examples, the procedure is applied to one-dimensional $L$, including and improving earlier results, and to two and three-dimensional $L$, providing new superintegrable systems.