The second order symmetry operators that commute with the Dirac operator with external vector, scalar and pseudo-scalar potentials are computed on a general two-dimensional spin-manifold. It is shown that the operator is defined in terms of Killing v
ectors, valence two Killing tensors and scalar fields defined on the background manifold. The commuting operator that arises from a non-trivial Killing tensor is determined with respect to the associated system of Liouville coordinates and compared to the the second order operator that arises from that obtained from the unique separation scheme associated with such operators. It shown by the study of several examples that the operators arising from these two approaches coincide.
Given an n-dimensional natural Hamiltonian L on a Riemannian or pseudo-Riemannian manifold, we call extension of L the n+1 dimensional Hamiltonian $H=frac 12 p_u^2+alpha(u)L+beta(u)$ with new canonically conjugated coordinates $(u,p_u)$. For suitable
L, the functions $alpha$ and $beta$ can be chosen depending on any natural number m such that H admits an extra polynomial first integral in the momenta of degree m, explicitly determined in the form of the m-th power of a differential operator applied to a certain function of coordinates and momenta. In particular, if L is maximally superintegrable (MS) then H is MS also. Therefore, the extension procedure allows the creation of new superintegrable systems from old ones. For m=2, the extra first integral generated by the extension procedure determines a second-order symmetry operator of a Laplace-Beltrami quantization of H, modified by taking in account the curvature of the configuration manifold. The extension procedure can be applied to several Hamiltonian systems, including the three-body Calogero and Wolfes systems (without harmonic term), the Tremblay-Turbiner-Winternitz system and n-dimensional anisotropic harmonic oscillators. We propose here a short review of the known results of the theory and some previews of new ones.
A procedure to extend a superintegrable system into a new superintegrable one is systematically tested for the known systems on $mathbb E^2$ and $mathbb S^2$ and for a family of systems defined on constant curvature manifolds. The procedure results e
ffective in many cases including Tremblay-Turbiner-Winternitz and three-particle Calogero systems.
