\(\lambda \) -- symmetries on the derivation of first integrals of ordinary differential equations (Q2884777)

For a second-order ODE, NEWLINE\[NEWLINE\ddot{u}=\phi (x, u, \dot{u}),NEWLINE\]NEWLINE let \(A=\partial _x+\dot{u}\partial _u+\phi (x, u, \dot{u})\partial _{\dot{u}}\) be the associated vector field on \(M^{(1)}=\{(x, u, \dot{u}); (x, u)\in M\},\) where \(M\) is an open set in \(X\times U\). Let \(v\) be a vector field on \(M\) and \(\lambda \in C^{\infty }(M^{(1)})\). On the set of pairs \((v, \lambda )\) an equivalence relation, called \(A\)-\textit{equivalence}, is introduced and it is proved that in the equivalence class of \((v, \lambda )\) there is an unique pair \((\partial _u, \bar{\lambda })\) called \textit{the canonical representative}. The authors show then that the canonical representative defines a \(\lambda \)-symmetry for \(A\) and two generators of the symmetry algebra of \(A\) can be used to obtain functionally independent first integrals if and only if they are in different \(A\)-equivalence classes.NEWLINENEWLINEFor the entire collection see [Zbl 1237.35006].