  In this article a study was made of the conditions under which a Hamiltonian
which is an element of the complex $ \left\{ h (1) \oplus h(1) \right\} \uplus
u(2) $ Lie algebra admits ladder operators which are also elements of this
algebra. The algebra eigenstates of the lowering operator constructed in this
way are computed and from them both the energy spectrum and the energy
eigenstates of this Hamiltonian are generated in the usual way with the help of
the corresponding raising operator. Thus, several families of generalized
Hamiltonian systems are found, which, under a suitable similarity
transformation, reduce to a basic set of systems, among which we find the 1:1,
2:1, 1:2, $su(2)$ and some other non-commensurate and commensurate anisotropic
2D quantum oscillator systems. Explicit expressions for the normalized
eigenstates of the Hamiltonian and its associated lowering operator are given,
which show the classical structure of two-mode separable and non-separable
generalized coherent and squeezed states. Finally, based on all the above
results, a proposal for new ladder operators for the $p:q$ coprime commensurate
anisotropic quantum oscillator is made, which leads us to a class of Chen
$SU(2)$ coherent states.
