  In recent years, many natural Hamiltonian systems, classical and quantum,
with constants of motion of high degree, or symmetry operators of high order,
have been found and studied. Most of these Hamiltonians, in the classical case,
can be included in the family of extended Hamiltonians, geometrically
characterized by the structure of warped manifold of their configuration
manifold. For the extended manifolds, the characteristic constants of motion of
high degree are polynomial in the momenta of determined form. We consider here
a different form of the constants of motion, based on the factorization
procedure developed by S. Kuru, J. Negro and others. We show that an important
subclass of the extended Hamiltonians admits factorized constants of motion and
we determine their expression. The classical constants may be non-polynomial in
the momenta, but the factorization procedure allows, in a type of extended
Hamiltonians, their quantization via shift and ladder operators, for systems of
any finite dimension.
