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In the three-dimensional flat space, a classical Hamiltonian, which has fivefunctionally independent integrals of motion, including the Hamiltonian, ischaracterized as superintegrable. Kalnins, Kress and Miller (J. Math. Phys. 48(2007), 113518, 26 pages) have proved that, in the case of nondegeneratepotentials, i.e. potentials depending linearly on four parameters, withquadratic symmetries, posses a sixth quadratic integral, which is linearlyindependent of the other integrals. The existence of this sixth integral implythat the integrals of motion form a ternary quadratic Poisson algebra with fivegenerators. The superintegrability of the generalized Kepler-Coulomb potentialthat was investigated by Verrier and Evans (J. Math. Phys. 49 (2008), 022902, 8pages) is a special case of superintegrable system, having two independentintegrals of motion of fourth order among the remaining quadratic ones. Thecorresponding Poisson algebra of integrals is a quadratic one, having the samespecial form, characteristic to the nondegenerate case of systems withquadratic integrals. In this paper, the ternary quadratic associative algebracorresponding to the quantum Verrier-Evans system is discussed. The subalgebrasstructure, the Casimir operators and the the finite-dimensional representationof this algebra are studied and the energy eigenvalues of the nondegenerateKepler-Coulomb are calculated.

Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential