Exact solutions of the Lane–Emden-type equation

We classify the Lane–Emden-type equation xy″+ny′+xνf(y)=0xy″+ny′+xνf(y)=0 with respect to the standard Lagrangian L=12xny′2-xn+ν-1∫f(y)dy according to the Noether point symmetries it admits. First integrals of the various cases, which admit Noether point symmetries, and reduction to quadratures for these cases are obtained. Six cases result in new solutions.