Exact solutions of generalized Lane-Emden equations of the second kind

Contact and Lie point symmetries of a certain class of second order differential equations using the Lie symmetry theory are obtained. Generators of these symmetries are used to obtain first integrals and exact solutions of the equations. This class of equations is transformed into the so-called generalized Lane-Emden equations of the second kind

\(y^{\prime\prime}(x)+{k\over{x}}y^{\prime}(x)+ g(x){\rm {e}}^{ny}=0.\) $$y^{\mathrm{\prime }\mathrm{\prime }}(x)+\frac{k}{x}{y}^{\mathrm{\prime }}(x)+g(x){\mathrm{e}}^{ny}=0.$$

Then we consider two types of functions g(x) and present first integrals and exact solutions of the Lane-Emden equation for them. One of the considered cases is new.