  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.
