Optimal system of subalgebras for sum of two ideals $\mathfrak{aff}(\mathbb{R})\oplus \mathfrak{sl}(2,\mathbb{R})$

Optimal system of subalgebras for one algebra Lie is constructed. This algebra Lie is direct sum of algebra of affine transformations group $Aff(\mathbb{R})$ and algebra of projective transformations group $SL(2,\mathbb{R})$. Some invariant solutions for one nonlinear partial differential equation are found.