  In this work the notion of Hamiltonian chain is presented as applied to
anisotropic oscillator potentials especially defined on three and four
dimensional Euclidean spaces. A Hamiltonian chain is a sequence of
superintegrable Hamiltonians which, in addition, constitute integrals of motion
of a new superintegrable system. Along with the Poisson algebras the quantum
counterparts of the chain is given as well as the eigenvalues of each member of
the chain. The method can be extended to cover n dimensional cases.
