Representations of algebra $sl_2(\mathbb R)$ and ordinary differential equations

We describe all nonequivalent representations of the algebra $sl_2(\mathbb{R})$ in the space of vector fields $\mathrm{Vect}\, \mathbb{R}^{2}$. For each of these representations all ordinary differential equations admitting representation data were found in terms of a basis differential invariants and operators of the invariant differentiation. We also found the Casimir operators of the corresponding universal enveloping algebra, the equations generated by the Casimir operator are integrated and the algebraic independence of the operators of invariant differentiation and Casimir operator are proved.