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We extend the method for constructing symmetry operators of higher order fortwo-dimensional quantum Hamiltonians by Kalnins, Kress and Miller (2010). Thisexpansion method expresses the integral in a finite power series in terms oflower degree integrals so as to exhibit it as a first-order differentialoperators. One advantage of this approach is that it does not require the apriori knowledge of the explicit eigenfunctions of the Hamiltonian nor theaction of their raising and lowering operators as in their recurrence approach(2011). We obtain insight into the two-dimensional Hamiltonians of radialoscillator type with general second-order differential operators for theangular variable. We then re-examine the Hamiltonian of Tremblay, Turbiner andWinternitz (2009) as well as a deformation discovered by Post, Vinet andZhedanov (2011) which possesses reflection operators. We will extend theanalysis to spaces of constant curvature. We present explicit formulas for theintegrals and the symmetry algebra, the Casimir invariant and oscillatorrealisations with finite-dimensional irreps which fill a gap in the literature.

Super-integrability and Deformed Oscillator Realisations of Quantum TTW Hamiltonians on Constant-Curvature Manifolds and with Reflections in a Plane