The Liouville equation and exactly transitive representations of algebra $sl_2(\mathbb{R})$

It is proved that exactly transitive representations of the algebra $sl_2(\mathbb{R})$ in the space of vector fields $\mathrm{Vect}\, \mathbb{R}^{3}$ are classified by solutions of the equation Liouville. We also obtain a characterization of exactly transitive representations of the algebra $so_3(\mathbb{R})$.