Generating functions and existence of contact symmetries of third order scalar ordinary differential equations

Analyzing the structure of generating functions associated to the contact symmetry condition of third order scalar ordinary differential equations, constraints guaranteeing that the ODE does not admit contact symmetries are obtained. Further, inspecting the subalgebras generated by contact symmetries having a specific form, an alternative procedure to generate third order equations with irreducible contact symmetry algebra sp(4)sp(4) is developed.