  We present a useful proposition for discovering extended Laplace-Runge-Lentz
vectors of certain quantum mechanical systems. We propose a new family of
superintegrable systems and construct their integrals of motion. We solve these
systems via separation of variables in spherical coordinates and obtain their
exact energy eigenvalues and the corresponding eigenfunctions. We give the
quadratic algebra relations satisfied by the integrals of motion. Remarkably
these algebra relations involve the Casimir operators of certain higher rank
Lie algebras in the structure constants.
