The Algebraic Structure of the First Integrals of Third-Order Linear Equations

We use the Lie theory of extended groups to analyse the first integrals of scalar third-order linear ordinary differential equations. The analysis reveals three natural classes for which the equation has four, five, or seven symmetries. We show that each class has first integrals with a specific number of symmetries. Illustrative examples of each class are given. Comparison is made with the integrals of nonlinear equations.