Extended Hamiltonians and shift, ladder functions and operators

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 Hamiltonians, 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 for systems of two degrees of freedom by S. Kuru, J. Negro and others. We show that an important subclass of the extended Hamiltonians, with arbitrary dimension, admits factorized constants of motion and we determine their expression. The classical constants can be polynomial or 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.