New inflationary exact solution from Lie symmetries

For the inflaton field we determine a new exact solution by using Lie symmetry analysis. Specifically, construct second-order differential master equation for arbitrary scalar potential assuming that spectral index density perturbations $n_{s}$ and to tensor ratio $r$ are related as $n_{s}-1=h\left( r\right) $. Function $h\left( $ is classified according admitted symmetries equation. The possible form $A_{2}$, $A_{3,2}$, $A_{3,3}$ $sl\left( 3,R\right) algebras. inflationary recovered of algebra. Scalar derived explicitly, while compare resulting indices with observations.