  In previous papers we determined necessary and sufficient conditions for the
existence of a class of natural Hamiltonians with non-trivial first integrals
of arbitrarily high degree in the momenta. Such Hamiltonians were characterized
as (n+1)-dimensional extensions of n-dimensional Hamiltonians on
constant-curvature (pseudo-)Riemannian manifolds Q. In this paper, we
generalize that approach in various directions, we obtain an explicit
expression for the first integrals, holding on the more general case of
Hamiltonians on Poisson manifolds, and show how the construction of above is
made possible by the existence on Q of particular conformal Killing tensors or,
equivalently, particular conformal master symmetries of the geodesic equations.
Finally, we consider the problem of Laplace-Beltrami quantization of these
first integrals when they are of second-degree.
