  In the three-dimensional flat space, a classical Hamiltonian, which has five
functionally independent integrals of motion, including the Hamiltonian, is
characterized as superintegrable. Kalnins, Kress and Miller (J. Math. Phys. 48
(2007), 113518, 26 pages) have proved that, in the case of nondegenerate
potentials, i.e. potentials depending linearly on four parameters, with
quadratic symmetries, posses a sixth quadratic integral, which is linearly
independent of the other integrals. The existence of this sixth integral imply
that the integrals of motion form a ternary quadratic Poisson algebra with five
generators. The superintegrability of the generalized Kepler-Coulomb potential
that was investigated by Verrier and Evans (J. Math. Phys. 49 (2008), 022902, 8
pages) is a special case of superintegrable system, having two independent
integrals of motion of fourth order among the remaining quadratic ones. The
corresponding Poisson algebra of integrals is a quadratic one, having the same
special form, characteristic to the nondegenerate case of systems with
quadratic integrals. In this paper, the ternary quadratic associative algebra
corresponding to the quantum Verrier-Evans system is discussed. The subalgebras
structure, the Casimir operators and the the finite-dimensional representation
of this algebra are studied and the energy eigenvalues of the nondegenerate
Kepler-Coulomb are calculated.
