In this paper we start with proving that the Schrödinger equation (SE) with the classical 12 -6 Lennard-Jones (L-J) potential is nonintegrable in the sense of the differential Galois theory (DGT), for any value of energy; i.e., there are no solutions in closed form for such differential equation.We study the 10-6 potential through DGT and SUSYQM; being it one of the two partner potentials built with a superpotential of the form w(r) ∝ 1/r 5 .We also find that it is integrable in the sense of DGT for zero energy.A first analysis of the applicability and physical consequences of the model is carried out in terms of the so called De Boer principle of corresponding states.A comparison of the second virial coefficient B(T ) for both potentials shows a good agreement for low temperatures.As a consequence of these results we propose the 10 -6 potential as an integrable alternative to be applied in further studies instead of the original 12 -6 L-J potential.Finally we study through DGT and SUSYQM the integrability of the SE with a generalized (2ν -2) -ν L-J potential.This analysis do not include the study of square integrable wave functions, excited states and energies different than zero for the generalization of L-J potentials.