The focus of this work is on the Lie and Noether symmetries of an interesting fourth-order Euler-Bernoulli beam ordinary differential equation or what the authors call a ‘fourth-order Emden-Fowler equation’. Firstly, a Lie group classification is performed and the results are tabulated. Then those symmetries which are Noether are obtained as well as the corresponding first integrals. Thereafter, invariant solutions are constructed. Finally, the authors deduce exact solutions to the Lane-Emden system which reduces to the original beam problem.