Contact symmetry algebras of scalar ordinary differential equations
The authors investigate third-order ordinary differential equations with respect to existence of irreducible contact symmetries and show that such an equation admits an irreducible contact symmetry algebra if and only if this equation may be transformed by a contact transformation into the equation \(y'''=0\). Furthermore, the classification of third-order equations by point symmetries is related to the question to interpret the corresponding Lie algebras of point symmetries, resp. of contact symmetries, as subalgebras of the symmetry algebras in the cases of maximal dimension (7 point symmetries, resp. 10 contactsymmetries) and of symmetry breaking. It is also proved that a fourth-order equation can not admit an irreducible contact symmetry algebra, for the order \(n\geq 5\), ODEs with nontrivial contact symmetry algebras are characterized.
scientific article; zbMATH DE number 1784468