  We generalise the construction of integrals of motion for quantum
superintegrable models and the deformed oscillator algebra approach. This is
presented in the context of 1D systems admitting ladder operators satisfying a
parabosonic algebra involving reflection operators and more generally
$c_{\lambda}$ extended oscillator algebras with grading. We apply the
construction on two-dimensional $c_{\lambda}$ oscillators. We also introduce
two new superintegrable Hamiltonians that are the anisotropic Dunkl and the
singular Dunkl oscillators. We construct the integrals and using this extended
approach of the Daskaloyannis method with grading and we present an algebraic
derivation of the energy spectrum of the two models from the finite dimensional
unitary representations and show how their spectrum divides into different
sectors and relates to the physical spectrum.
