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.