We show that the quadratic symmetry algebra  of the two dimensional free fall problem is related to the so-called W. H. Klink quartic algebra of the quartic anharmonic oscillator. Through this correspondence,newintegralpropertiesofthe oscillator eigenfunctions are established. In particular, the existence  of an infinite number of integral operators, mutually commuting and commuting with the quartic anharmonic oscillator Hamiltonian is proven.

special functions, quantum anharmonic oscillators, symmetry groups.