  We extend the method for constructing symmetry operators of higher order for
two-dimensional quantum Hamiltonians by Kalnins, Kress and Miller (2010). This
expansion method expresses the integral in a finite power series in terms of
lower degree integrals so as to exhibit it as a first-order differential
operators. One advantage of this approach is that it does not require the a
priori knowledge of the explicit eigenfunctions of the Hamiltonian nor the
action of their raising and lowering operators as in their recurrence approach
(2011). We obtain insight into the two-dimensional Hamiltonians of radial
oscillator type with general second-order differential operators for the
angular variable. We then re-examine the Hamiltonian of Tremblay, Turbiner and
Winternitz (2009) as well as a deformation discovered by Post, Vinet and
Zhedanov (2011) which possesses reflection operators. We will extend the
analysis to spaces of constant curvature. We present explicit formulas for the
integrals and the symmetry algebra, the Casimir invariant and oscillator
realisations with finite-dimensional irreps which fill a gap in the literature.
