Integration of third order ordinary differential equations by Lie's method: equations admitting three-dimensional Lie algebras

We show how to integrate third order differential equations which admit a three-dimensional symmetry Lie algebra L3. If L3 is solvable, then we integrate the equation by quadrature, in accordance with Lie’s theory. If L3 is not solvable, then we can still integrate the given third order equation by reducing it to a first order equation, which can be transformed into a Riccati equation, thanks to the fact that L3 transforms one of the symmetry of the third order equation into a non-local symmetry of the first order equation. Some examples are provided.