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 $2y'y'''- 3{y''}^2= 0$ is found to have six symmetries, while its first integrals have three, two and one symmetry, respectively. The equation $(1+{y'}^2)y'''=(3y'+1){y''}^2$ 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.

Reviewer: Maria Clara Nucci (MR 96e:34013)