Here, the authors compare the Lie point symmetries of linear third-order ordinary differential equations with the Lie point symmetries of their linear second-order first integrals.
In particular, they find that if the third-order equation has seven symmetries, then two of its first integrals have four symmetries, and one has three; if the third-order equation has five symmetries, then all three first integrals have three symmetries; if the third-order equation has four symmetries, then all three first integrals have two symmetries. The authors do not give a theoretical explanation of such findings.
Finally, the authors show two examples of nonlinear third-order equations. The equation is found to have six symmetries, while its first integrals have three, two and one symmetry, respectively. The equation is found to have four symmetries, while one of its first integrals has three symmetries, and two have one symmetry. No general rule is provided for nonlinear equations.