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We present a useful proposition for discovering extended Laplace-Runge-Lentzvectors of certain quantum mechanical systems. We propose a new family ofsuperintegrable systems and construct their integrals of motion. We solve thesesystems via separation of variables in spherical coordinates and obtain theirexact energy eigenvalues and the corresponding eigenfunctions. We give thequadratic algebra relations satisfied by the integrals of motion. Remarkablythese algebra relations involve the Casimir operators of certain higher rankLie algebras in the structure constants.

Extended Laplace-Runge-Lentz vectors, new family of superintegrable systems and quadratic algebras