Families of 2D superintegrable anisotropic Dunkl oscillators and algebraic derivation of their spectrum

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.