[en] We study a pair of canonoid (fouled) Hamiltonians of the harmonic oscillator which provide, at the classical level, the same equation of motion as the conventional Hamiltonian. These Hamiltonians, say K₁ and K₂, result to be explicitly time dependent and can be expressed as a formal rotation of two cubic polynomial functions, H₁ and H₂, of the canonical variables (q,p). We investigate the role of these fouled Hamiltonians at the quantum level. Adopting a canonical quantization procedure, we construct some quantum models and analyze the related eigenvalue equations. One of these models is described by a Hamiltonian admitting infinite self-adjoint extensions, each of them has a discrete spectrum on the real line. A self-adjoint extension is fixed by choosing the spectral parameter ε of the associated eigenvalue equation equal to zero. The spectral problem is discussed in the context of three different representations. For ε=0, the eigenvalue equation is exactly solved in all these representations, in which square-integrable solutions are explicitly found. A set of constants of motion corresponding to these quantum models is also obtained. Furthermore, the algebraic structure underlying the quantum models is explored. This turns out to be a nonlinear (quadratic) algebra, which could be applied for the determination of approximate solutions to the eigenvalue equations