Symmetries of first integrals and their associated differential equations (Q1300019)

The authors show by various examples that the concepts of finding first integrals and reducing the order of an ODE by symmetries are fundamentally different. From their abstract: ``The relationship between the reduction of order through point symmetries and integration is explored with particular emphasis on the lost and gain of point (contact for third order) symmetries to and from nonlocal symmetries.'' Especially, the third-order equation \(y'''=0\) is investigated with respect to first integrals related to the ten contact symmetries and reduction of order with respect to special symmetries.