Lie group classification of the generalized Lane–Emden equation

We carry out the Lie group classification of the generalized Lane–Emden equation xu″+nu′+xH(u)=0xu″+nu′+xH(u)=0, which has many applications in mathematical physics and astrophysics. We show that the equation admits a three-dimensional equivalence Lie algebra. It is also shown that the principal Lie algebra, which in this case is trivial, has seven possible extensions. Three new cases arise for which the Lie point symmetry algebra is non-trivial. Comparison is then made of these cases with the Noether symmetry cases as well as the partial Noether operators.