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Generalized Solvable Structures and First Integrals for ODEs Admitting an Symmetry Algebra

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.

In this paper we focus on the concept of solvable structure, which provides a useful tool for integrating by quadratures ODEs lacking of local symmetries or admitting a nonsolvable symmetry algebra [23]. Although, under some regularity assumptions, the local existence of a solvable structure for any given ODE is guaranteed, its explicit determination is, in general, a quite difficult task. A noteworthy simplification may be obtained by looking for solvable structures which are adapted to some admitted symmetry algebra. For example, in [9,10] solvable structures adapted to local and nonlocal symmetry algebras are considered, while in [24] a solvable structure for any third-order ODE admitting a nonsolvable symmetry algebra isomorphic to sl(2, R) is constructed exploiting the symmetry generators. In this paper we address a generalization of the results of [24], considering ODEs of arbitrary order k > 3 admitting a nonsolvable symmetry algebra isomorphic to sl (2, R). For this class of equations we introduce the notion of generalized solvable structure and we prove that the knowledge of a generalized solvable structure allows the construction of a complete set of first integrals for the equation, given in terms of two independent solutions to a corresponding second-order linear ODE. As a consequence, the general solution to the equation can be expressed in parametric form in terms of a fundamental set of solutions to a (k − 3)-parameter family of second-order linear equations. It is important to remark that, despite the symmetry algebra is nonsolvable, the first integrals can be computed by quadratures. Moreover, the generators of the symmetry algebra sl(2, R) and the vector fields belonging to the generalized solvable structure do not form a (standard) solvable structure.
The paper is organized as follows. We start Section 2 briefly recalling the main definitions and facts about distributions of vector fields and their symmetries and the concept of solvable structure, as well as its role in the integration by quadratures of ODEs. Afterwards we define generalized solvable structures for kth-order ODEs admitting an sl(2, R) symmetry algebra and we prove a result which allows us to exploit the vector fields belonging to the generalized solvable structure for computing by quadratures k − 3 functionally independent first integrals shared by the equation and the symmetry generators of sl(2, R). In particular, it is proved that the restriction of the ODE to the generic leaf defined by these first integrals provides a third-order equation which inherits sl(2, R) as Lie symmetry algebra. Since the classical Lie reduction of order for this kind of equations ends with a Riccati-type equation that does not permit to recover a closed form expression for the general solution to the third-order ODE [5,17]. We exploit the recent results on the first integrals and general solutions to this class of equations [24,25], in order to complete the set of first integrals. In particular we obtain the remaining three functionally independent first integrals in terms of the solutions to a related second-order ODE. The complete set of k independent first integrals can be used to provide the general solution to the original kth-order equation, expressed, in explicit or parametric form, in terms of a fundamental set of solutions to a (k − 3)-parameter family of second-order linear equations.
In Section 3 our results are applied to fourth and fifth order ODEs, whose symmetry algebra is three-dimensional and isomorphic to sl(2, R). Since the symmetry algebra is nonsolvable and its dimension is strictly lower than the order of the ODE, the integrability by quadratures cannot be guaranteed by the classical Lie theory. For each one of the two examples of fourth-order equations, the construction of a generalized solvable structure requires the determination of a single vector field, and allows us to provide the general solutions to the ODEs (in explicit and parametric form, respectively) in terms of the solutions to a one-parameter family of second-order linear equations. For the fifth-order ODE presented in Section 3.3, the generalized solvable structure is formed by two vector fields and the general solution to the equation is expressed in parametric form in terms of the solutions to a two-parameter family of Schrödinger-type equations.
an open domain of N where the vector fields {A 1 , . . . , A n−k } are pointwise linearly independent, we say that A is a distribution of maximal rank n − k (or of codimension k) on U. It is well known that any integrable distribution A of maximal rank on U ⊆ N determines a (n − k)-dimensional foliation of U. If this foliation is described through the vanishing of k functions of the form I h − c h , where I h ∈ C ∞ (U) and c h ∈ R, we can chose dI 1 , . . . , dI k as generators for the Pfaffian system annihilating the distribution A . A submanifold S ⊂ N is an integral manifold for A if A | S ⊆ T S. If in particular A | S = T S we say that S is a maximal integral manifold for A .
Given a distribution A , a vector field X is a symmetry of A if [X, A] ∈ A , for any A ∈ A . Let A and B be two distributions on N. We say that A and B are transversal at p ∈ N if they do not vanish at p and A (p) ∩ B(p) = {0}. Analogously, A and B are transversal in U if they are transversal at any point of U. An algebra G of symmetries for a distribution A is nontrivial if G generates a distribution which is transversal to A .

Solvable structures for ODEs
It is well known that, given a (n − k)-dimensional integrable distribution A on an n-dimensional manifold N, the knowledge of a solvable k-dimensional algebra G of nontrivial symmetries for A guarantees that a complete set of first integrals for A can be found by quadratures. Solvable structures provide an extension of this classical result, significantly enlarging the class of vector fields which can be used to integrate by quadratures a distribution of vector fields.
In this section we recall some basic definitions and facts on solvable structures. The interested reader is referred to [6,8,16,26] for further details.
, the following conditions hold: . . , A n−k be an integrable (n − k)-dimensional distribution defined on an orientable n-dimensional manifold N and let {Y 1 , . . . ,Y k } be a solvable structure for A . Denoting by Ω a volume form on N and by α the k-form A 1 . . . A n−k Ω, the distribution A can be described as the annihilator of the Pfaffian system generated by where the hat denotes omission of the corresponding vector field and ∆ is the function on N defined by Moreover, the forms ω i satisfy Therefore the integral manifolds of the distribution A can be described in implicit form as the level manifolds Proof. The interested reader is referred to the original papers [6-8, 16, 26] for a proof of this theorem.
We remark that, if the distribution A admits an Abelian Lie algebra of symmetries generated by the vector fields Y 1 , . . . ,Y k , then all the 1-forms ω i are closed, i.e. the function 1/∆ provides an integrating factor for all the 1-forms The main difference between a solvable structure and a solvable symmetry algebra for a completely integrable distribution A is that the fields belonging to a solvable structure do not need to be symmetries of A . This, of course, gives more freedom in the choice of the vector fields which can be exploited to find integral manifolds of A by quadratures.
Let (x, u (k−1) ) = (x, u, u 1 , . . . , u k−1 ) denote the coordinates of the jet space J k−1 (R, R), for k ≥ 2. In order to apply Theorem 2.1 to the integration of ODEs, we recall that, with any kth-order ODE in normal form we can associate the vector field A defined on a suitable domain U ⊂ J k−1 (R, R) and given by If E is a kth-order ODE of the form (2.2) and A is the corresponding vector field given by (2.3), the vector fields Corollary 2.1. The knowledge of a k-dimensional solvable structure for a kth-order ODE, allows us to obtain the solution to the ODE by quadratures.
Proof. The distribution A is obviously integrable, being one-dimensional. Moreover the functions I h of Theorem 2.1 provide a complete set of first integrals for A.

Generalized solvable structures for ODEs
In this section we generalize the notion of solvable structure for ODEs, in order to exploit the knowledge of a symmetry algebra isomorphic to sl(2, R) for kth-order ODEs with k > 3. In particular we give the following Definition 2.3. Let E be a kth-order ODE of the form (2.2) with k > 3 and let A be the corresponding vector field defined by (2.3). If E admits a symmetry algebra isomorphic to sl(2, R) with gener- We remark that the k vector fields {X 1 , X 2 , X 3 ,Y 1 , . . . ,Y k−3 } do not form a solvable structure for the distribution generated by A, due to the commutation relations between X i . Despite this fact we have the following result.
Theorem 2.2. Let E be a kth-order ODE with k > 3 admitting a symmetry algebra isomorphic to sl(2, R) and let A be the corresponding vector field defined by (2.3). The knowledge of a generalized solvable structure {Y 1 , . . . ,Y k−3 } for E allows us to construct k first integrals for the ODE (2.2) in terms of two independent solutions to a linear second order ODE.
Proof. Since {Y 1 , . . . ,Y k−3 } provide a solvable structure for the integrable distribution A = A, X 1 , X 2 , X 3 , we can use Theorem 2.1 in order to find suitable functions {I 1 , . . . , Moreover, since the vector fields A, X 1 , X 2 , X 3 are tangent to Σ, equation (2.2) restricted to Σ turns out to be a third-order ODE admitting sl(2, R) as symmetry algebra. Therefore, we can exploit the results of [24,25] in order to obtain the remaining three first integrals. In particular, starting from the restriction of equation (2.2) to the 4-dimensional submanifold Σ, we get I k−2 , I k−1 and I k in terms of two dependent solutions to a second-order linear ODE and of the real constants c i , i = 1, . . . , k − 3.

Example 1
Let us consider the fourth-order equation which corresponds, when u 1 = 0, to the vector field The symmetry algebra of equation (3.1) is spanned by and hence it is three-dimensional and isomorphic to sl(2, R). Since equation (3.1) lacks of further symmetries, we can look for a generalized solvable structure. In particular we have to find a vector field Y on a suitable domain U ⊂ J 3 (R, R) such that Y is a symmetry of the distribution A = A, X 1 , X 1 , X 3 , where X i stands for the third-order prolongation of v i , for i = 1, 2, 3, i.e.
It is worthwhile to remark that finding a symmetry for the distribution A = A, X 1 , X 1 , X 3 is definitely simpler than finding a symmetry for the vector field A. For instance, it is easy to check that the vector field provides a generalized solvable structure for ODE (3.1), due to the following commutation relations (3.5) Once we have got Y we can apply Theorem 2.2, by considering the volume form Ω = dx ∧ du ∧ du 1 ∧ du 2 ∧ du 3 and the differential 1-form α = A X 1 X 2 X 3 Ω given by Since Y is a symmetry of α, the function 1/∆, with is an integrating factor of α. Therefore, the one-form is closed and so (locally) exact, i.e. ω = dI, where the function I = I(x, u (3) ) is given (up to an additive constant) by: Function (3.6) provides a first integral for the integrable distribution A, X 1 , X 2 , X 3 and the restriction of (3.1) to the submanifold I = c 1 , where c 1 ∈ R, leads to the reduced equation Equation (3.7) inherits a symmetry algebra isomorphic to sl(2, R) and spanned by the restrictions of the vector fields (3.2) to I = c 1 . Moreover (3.7) corresponds to the equation presented in [25], Case 1 in Table 2, for the function C(u) = u 2 − c 1 . By Proposition 4.1 in [25], a complete set of first integrals for equation (3.7) can be expressed in terms of two independent solutions to the secondorder linear ODE (see Case 1, Table 6 in [25]): Let Ψ c 1 ;1 (u) and Ψ c 1 ;2 (u) denote two linearly independent solutions to (3.8), such that the corresponding Wronskian is equal to 1 (note that the Wronskian is constant by the Liouville's formula).
According to Table 7 (Case 1) in [25], three independent first integrals for equation (3.7) are given by , , in (3.9), the resulting functions I 1 , I 2 , I 3 provide, together with (3.6), a complete set of first integrals for equation (3.1). Therefore, the general solution to (3.1) is given by (see Section 5 in [25]): We finally remark that the solutions to (3.8) can be expressed in terms of Whittaker functions: if M µ,ν (z) and W µ,ν (z) denote the Whittaker functions of parameters µ = i 4 c 1 and ν = 1 4 , then two linearly independent solutions to (3.8) are given by Consequently, the general solution (3.10) can be expressed as follows:

Example 2
Let us consider the fourth-order ODE and let A denote the corresponding vector field of the form The Lie symmetry algebra of equation (3.11) is three-dimensional and isomorphic to sl(2, R). The third-order prolongations of the corresponding Lie symmetry generators are given by

Co-published by Atlantis Press and Taylor & Francis
Copyright: the authors A symmetry of the integrable distribution A, X 1 , X 2 , X 3 can be easily found searching, for instance, a symmetry of the form η(x, u (3) )∂ u . In particular, if we consider the vector field we have the following commutation relations for suitable functions µ i , ρ i (i = 0, 1, 2, 3) whose expressions are omitted, not being involved in the following discussion. Since the vector field (3.12) provides a generalized solvable structure for equation (3.11), an integrating factor for the 1-form α = A X 1 X 2 X 3 Ω is given by 1/∆, where and we assume that ∆ = 0. Therefore, the differential one-form ω = 1 ∆ α is closed and, locally, exact. A corresponding first integral is given by (3.14) The restriction of equation (3.11) to a generic leaf I 1 = c 1 , c 1 ∈ R, provides the following third-order ODE: (3.15) Equation (3.15) inherits the symmetry algebra sl(2, R) and corresponds to the third-order ODE appearing in [25, Case 3 in Table 1], for the particular case of According to [25], a complete set of first integrals to (3.15) can be expressed in terms of a system of solutions to the second-order linear equation In particular, if Ψ c 1 ;1 = Ψ c 1 ;1 (s), Ψ c 1 ;2 = Ψ c 1 ;2 (s) denote two linearly independent solutions to (3.16) and W = W (Ψ c 1 ;1 , Ψ c 1 ;2 )(s) denotes the corresponding Wronskian, a complete system of first integrals for equation (3.15) is given by (see [25,Case 3 in Table 8]): √ u 1 Ψ c 1 ;2 (s) + u(3s 2 + 4c 1 − 24)Ψ c 1 ;2 (s) , (3.17) By setting J i = J i (x, u (3 )) = I i (x, u, u 1 , u 2 ; I 1 ), for i = 2, 3, 4, we finally obtain that {I 1 , J 2 , J 3 , J 4 } is a complete set of first integrals for the fourth-order equation (3.11). Hence, the general solution to (3.15), and therefore to the original equation (3.11), is implicitly defined by where c i ∈ R for i = 1, 2, 3, 4.
In order to obtain a parametric expression for the implicit solution (3.18), we consider a new parameter t such that s = s(t) is determined as follows (see [25], Section 5 for further details): = 3s(t) 2 + 4c 1 − 24. • If c 1 = 6 then a solution to (3.19) is locally given by and by means of the transformation φ c 1 (t) = Ψ c 1 (s(t)), equation (3.16) is mapped into For each one of the three cases considered above, let φ 1;c 1 (t) = Ψ c 1 ;1 (s(t)) and φ 2;c 1 (t) = Ψ c 1 ;2 (s(t)) denote two linearly independent solutions to the second-order linear equations (3.20), (3.21), and (3.22), respectively. In terms of these functions, the general solution to equation (3.11) can be expressed in parametric as follows [25]: (3.23)

Example 3
In this section we consider the fifth-order ODE on a suitable domain U ⊂ J 5 (R, R) such that uu 3 = 0 and its associated vector field The Lie symmetry algebra of equation (3.24) is three dimensional and isomorphic to sl(2, R). The corresponding fourth-order prolongations of the symmetry generators are A symmetry Y 1 of the integrable distribution A, X 1 , X 2 , X 3 can be easily determined by searching, for instance, a vector field of the form η(x, u (4) )∂ u 2 . For example, the vector field satisfies the following commutation relations: (3.25) The next step in order to obtain a generalized solvable structure for equation (3.24) is looking for a symmetry Y 2 of the integrable distribution A, X 1 , X 2 , X 3 ,Y 1 . It is easy to check that the vector field Co-published by Atlantis Press and Taylor & Francis Copyright: the authors three functionally independent first integrals for equation (3.28) are given by (3.30) In order to obtain a parametric general solution from (3.30), we consider a new parameter t such that s = s(t) is determined as follows: s (t) = 1 C(s(t)) = 4 sin(2c 1 − s(t)) √ c 2 , which yields s(t) = − arctan 2e −4t/ √ c 2 e −8t/ √ c 2 − 1 + 2c 1 . If φ 1 = φ 1 (t; c 1 , c 2 ) and φ 2 = φ 2 (t; c 1 , c 2 ) are two linearly independent solutions to (3.32) such that W (φ 1 , φ 2 )(t; c 1 , c 2 ) = 1, then the parametric general solution to equation (3.24) can be expressed as follows

Conclusions
Generalized solvable structures for ODEs of order k > 3 admitting a symmetry algebra isomorphic to sl(2, R) have been introduced, providing a powerful tool for the explicit determination of the general solutions to these equations. A generalized solvable structure is formed by k − 3 vector fields, which, in general, are not symmetries of the equation. Moreover, the symmetry generators of sl(2, R) and the vector fields of the generalized solvable structure do not form a solvable structure. Nevertheless, the knowledge of a generalized solvable structure allows the determination by quadratures of a complete set of first integrals, and hence of the general solution to the equation, in terms of two independent solutions to an associated family of second-order linear ODEs. This link between equations admitting sl(2, R) and second-order linear ODEs is, to the best of our knowledge, new in the literature and generalizes the recent results obtained for third-order equations.
The concept of generalized solvable structure gives more freedom in the choice of the vector fields which can be used to integrate the equation under study. The effectiveness of the proposed method is illustrated by several examples of equations admitting only an sl(2, R) algebra of Lie symmetries. In these cases, after finding a generalized solvable structure, we provide a complete set of first integrals as well as the general solutions in parametric or explicit forms.