*(English)*Zbl 0986.34033

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

##### MSC:

34C14 | Symmetries, invariants (ODE) |

34A30 | Linear ODE and systems, general |

34A25 | Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.) |