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We describe a procedure to construct polynomial in the momenta firstintegrals of arbitrarily high degree for natural Hamiltonians $H$ obtained asone-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 naturalintegrability conditions, it is necessary for the construction that thecurvature of the metric tensor associated with $L$ is constant. As examples,the procedure is applied to one-dimensional $L$, including and improvingearlier results, and to two and three-dimensional $L$, providing newsuperintegrable systems.

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