It is investigated how two (standard or generalized) $$\lambda$$-symmetries of a given second-order ordinary differential equation can be used to solve the equation by quadratures. The method is based on the construction of two commuting generalized symmetries for this equation by using both $$\lambda$$-symmetries. The functions used in that construction are related with integrating factors of the reduced and auxiliary equations associated to the $$\lambda$$-symmetries. These functions can also be used to derive a Jacobi last multiplier and two integrating factors for the given equation. Some examples illustrate the method; one of them is included in the XXVII case of the Painlevé-Gambier classification. An explicit expression of its general solution in terms of two fundamental sets of solutions for two related second-order linear equations is also obtained.

A complete set of first integrals for any third-order ordinary differential equation admitting a Lie symmetry algebra isomorphic to s l ( 2 , R ) is explicitly computed. These first integrals are derived from two linearly independent solutions of a linear second-order ODE, without additional integration. The general solution in parametric form can be obtained by using the computed first integrals. The study includes a parallel analysis of the four inequivalent realizations of s l ( 2 , R ) , and it is applied to several particular examples. These include the generalized Chazy equation, as well as an example of an equation which admits the most complicated of the four inequivalent realizations.

The notion of solvable structure is generalized in order to exploit the presence of an algebra of symmetries for a kth-order ordinary differential equation with k > 3. In this setting, the knowledge of a generalized solvable structure for allows us to reduce to a family of second-order linear ordinary differential equations depending on k − 3 parameters. Examples of explicit integration of fourth and fifth order equations are provided in order to illustrate the procedure.

For a system of ordinary differential equations (ODEs) or, more generally, an involutive distribution of vector fields, the problem of its integration is considered. Among the many approaches to this problem, solvable structures provide a systematic procedure of integration via Pfaffian equations that are integrable by quadratures. In this paper structures more general than solvable structures (named  ∞ -structures) are considered. The symmetry condition in the concept of solvable structure is weakened for  ∞ -structures by requiring their vector fields be just  ∞ -symmetries. For  ∞ -structures there is also an integration procedure, but the corresponding Pfaffian equations, although completely integrable, are not necessarily integrable by quadratures. The well-known result on the relationship between integrating factors and Lie point symmetries for first-order ODEs is generalized for  ∞ -structures and involutive distributions of arbitrary corank by introducing symmetrizing factors. The role of these symmetrizing factors on the integrability by quadratures of the Pfaffian equations associated with the  ∞ -structure is also established. Some examples that show how these objects and results can be applied in practice are also presented.

A novel procedure to reduce by four the order of Euler–Lagrange equations associated to n th order variational problems involving single variable integrals is presented. In preparation, a new formula for the commutator of two C ∞ -symmetries is established. The method is based on a pair of variational C ∞ -symmetries whose commutators satisfy a certain solvability condition. It allows one to recover a ( 2 n − 2 ) -parameter family of solutions for the original 2 n th order Euler–Lagrange equation by solving two successive first order ordinary differential equations from the solution of the reduced Euler–Lagrange equation. The procedure is illustrated by two different examples.