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In previous papers we determined necessary and sufficient conditions for theexistence of a class of natural Hamiltonians with non-trivial first integralsof arbitrarily high degree in the momenta. Such Hamiltonians were characterizedas (n+1)-dimensional extensions of n-dimensional Hamiltonians onconstant-curvature (pseudo-)Riemannian manifolds Q. In this paper, wegeneralize that approach in various directions, we obtain an explicitexpression for the first integrals, holding on the more general case ofHamiltonians on Poisson manifolds, and show how the construction of above ismade 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 thesefirst integrals when they are of second-degree.

Generalizations of a method for constructing first integrals of a class of natural Hamiltonians and some remarks about quantization