Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and Integrability

Abstract

We present a systematic procedure for the determination of a complete set of kth-order (𝑘≥2) differential invariants corresponding to vector fields in three variables for three-dimensional Lie algebras. In addition, we give a procedure for the construction of a system of two kth-order ODEs admitting three-dimensional Lie algebras from the associated complete set of invariants and show that there are 29 classes for the case of k = 2 and 31 classes for the case of 𝑘≥3. We discuss the singular invariant representations of canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras. Furthermore, we give an integration procedure for canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras which comprises of two approaches, namely, division into four types I, II, III, and IV and that of integrability of the invariant representations. We prove that if a system of two second-order ODEs has a three-dimensional solvable Lie algebra, then, its general solution can be obtained from a partially linear, partially coupled or reduced invariantly represented system of equations. A natural extension of this result is provided for a system of two kth-order (𝑘≥3) ODEs. We present illustrative examples of familiar integrable physical systems which admit three-dimensional Lie algebras such as the classical Kepler problem and the generalized Ermakov systems that give rise to closed trajectories.

1. Introduction

Realizations of Lie algebras in terms of vector fields are applied, in particular, for the integration and classification of ordinary differential equations (see, e.g. [1–5]). In spite of the importance for applications, the problem of complete description of realizations has not been solved even for cases when either the dimension of the algebras or the dimension of the realization space is a fixed small integer [6]. The classification of finite-dimensional Lie algebras was initiated by Lie [7] who also presented realizations in terms of vector fields in (1+1)- and (1+2)-dimensional spaces. In the case of vector fields in the two-dimensional space, he gave a complete classification in the complex domain. For realizations of vectors fields in the three-dimensional space, Lie presented a partial classification which was recently completed. All the possible complex Lie algebras of dimension less than four were already obtained by Lie himself [8, 9]. Bianchi [10] investigated three-dimensional real Lie algebras. Complete and correct classification of four-dimensional real Lie algebras was obtained by Mubarakzjanov [11]. Wafo Soh and Mahomed [12] used the results of Bianchi and Mubarakzjanov to classify realizations of three- and four-dimensional real Lie algebras in the space of three variables, and they also described systems of two second-order ODEs admitting real four-dimensional Lie algebras.

The realizations of real Lie algebras of dimension no greater than four of vector fields in the space of an arbitrary (finite) number of variables were given in [6]. Popovych et al. [6] used the results of Bianchi and Mubarakzjanov with a different approach. They found that the work [12] does not provide the complete list of realizations of real Lie algebras of dimensions three and four.

Systems of two second-order ODEs arise frequently in several applications in classical, fluid, and quantum mechanics as well as in general relativity. Hence, these have been studied vigorously over the years. A sample of the literature as far as symmetry Lie algebras of ODEs are concerned can be found in [13–15]. In particular, the classification of systems of two second-order ODEs admitting real four-dimensional Lie algebras was attempted in [12]. Gaponova and Nesterenko [16] gave a classification of system of two second-order ODEs admitting real three- and four-dimensional Lie algebras by using the results of [6].

In the this paper, by utilizing realizations given in [6], a systematic procedure for finding a complete set of 𝑘th-order (𝑘≥2) differential invariants including basis of invariants corresponding to vector fields in three variables for three-dimensional Lie algebras is presented. In addition, we give a procedure for the construction of two 𝑘th-order ODEs possessing three-dimensional Lie algebras from the associated complete set of invariants. We show that there are 29 classes for 𝑘=2 and 31 classes for 𝑘≥3. Moreover, we mention those cases in which regular systems of two second-order ODEs are not obtained for three-dimensional Lie algebras. We present a brief discussion on the singular representation of canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras. Furthermore, we provide an integration procedure for canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras. This procedure is composed of two approaches, namely, one which depends on general observations of canonical forms which admit three-dimensional Lie algebras; the second depends on a complete set of differential invariants and is applicable only for those cases which admit three-dimensional solvable Lie algebras. We show by familiar physical examples how these integration approaches work. In the second approach, for solvable Lie algebras, we analyze integrability in terms of integration of the invariant representations. We prove that if a system of two second-order ODEs admits a solvable three-dimensional Lie algebra, then, its general solution can be deduced from a partially linear, partially coupled or invariant reduced system of equations. A natural extension of this result is stated for a system of two 𝑘th-order (𝑘≥3) ODEs.

In Table 1, for ease of reference, we list the realizations of three-dimensional Lie algebras as given in [6]. In the second section, for the Lie algebras of vector fields considered, we discuss and deduce a complete set of differential invariants including basis of invariants with the construction of the corresponding systems of two 𝑘th-order ODEs. Moreover, for systems of two second-order ODEs admitting three-dimensional Lie algebras, we discuss the singular representation and those cases in which we do not obtain systems. In the end, in Table 2, we give a complete list of second-order differential invariants corresponding to vector fields in three variables of three-dimensional real Lie algebras and the associated complete list of canonical forms for systems of two second-order ODEs. In Section 3, we provide an integration procedure for the canonical forms of systems of two 𝑘th-order (𝑘≥2) ODEs admitting three-dimensional Lie algebras. This is stated as theorems for systems admitting solvable algebras. Reduction of canonical forms for systems of two second-order ODEs into invariant systems of first-order and algebraic equations are given in Table 2. The paper ends with a brief conclusion.

Table 1 

Realizations of three-dimensional Lie algebras in three variables.

Table 2 

Second-order differential invariants and equations.

2. Invariants and Systems of ODEs

When one calculates the symmetries of a given differential equation, one finds the generators in the form of vector fields and then computes the Lie brackets to get the structure constants of the particular Lie algebra one has found. We can start from a given Lie algebra with a set of structure constants and ask which vector fields in at most three variables satisfy the given Lie bracket relations with none of the vector fields vanishing. We thus ask for possible realizations of the given Lie algebra. We make use of the realizations of three-dimensional Lie algebras in the (1+2)-dimensional space given in [6]. We utilize these realizations in order to obtain differential invariants and invariant representations for system of two second-order and higher-order ODEs. Extensions to higher order invariants are given to understand the behavior of such invariants. This is the inverse problem in symmetry analysis and was initiated by Lie.

For completeness and ease of reference, we list the realizations of three-dimensional real algebras as given in [6] in Table 1. However, we mostly keep the format of [12].

Differential invariants play an important role in the structure and reduction of differential equations. Differential invariants can be obtained by various approaches and we refer, for example, to [16] for realizations in (1+2)-dimensional space. Differential invariants obtained by various approaches corresponding to three-dimensional Lie algebras have very interesting properties. In the investigation of the invariant approach for a system of two second-order ODEs admitting three-dimensional Lie algebras, it is found that, for a regular system of ODEs, there are four independent invariants, three of which play the role of basis and the fourth one is derived from the remaining three.

For the case of a regular system of two second-order ODEs admitting three-dimensional Lie algebras, there are four functionally independent invariants. For example, consider the system of two second-order ODEs 𝑥̈𝑥̇²𝑥̇=𝑓𝑦,𝑦̇,𝑥̈𝑦̇²−𝑥̇𝑦̈𝑦̇³𝑥̇=𝑔𝑦,𝑦̇, (1) admitting the three-dimensional Lie algebra 𝐴³_3,2. We have the four independent invariants 𝑤₁=𝑦,𝑤₂=𝑥̇𝑦̇,𝑤₃=𝑥̈𝑥̇²,𝑤₄=𝑥̈𝑦̇²−𝑥̇𝑦̈𝑦̇³. (2) Here, the invariants 𝑤₃ and 𝑤₄ are of order two, that is, equal to the order of the system of ODEs that is constructible. Similarly for the regular system of two third-order ODEs 𝑥⃛𝑥̇²𝑦̇−2𝑥̈²𝑥̇³𝑦̇𝑥̇=𝑓𝑦,𝑦̇,𝑥̈𝑥̇²,𝑥̈𝑦̇²−𝑥̇𝑦̈𝑦̇³,−𝑥̇𝑦⃛𝑦̇⁴+𝑥⃛𝑦̇²−3𝑥̈𝑦𝑦̇̈+3𝑥̇𝑦̈²𝑦̇⁵𝑥̇=𝑔𝑦,𝑦̇,𝑥̈𝑥̇²,𝑥̈𝑦̇²−𝑥̇𝑦̈𝑦̇³, (3) admitting the same Lie algebra 𝐴³_3,2; we have the six invariants𝑤₁=𝑦,𝑤₂=𝑥̇𝑦̇,𝑤₃=𝑥̈𝑥̇²,𝑤₄=𝑥̈𝑦̇²−𝑥̇𝑦̈𝑦̇³,𝑤₅=𝑥⃛𝑥̇²𝑦̇−2𝑥̈²𝑥̇³𝑦̇,𝑤₆𝑥̇𝑦⃛=−𝑦̇⁴+𝑥⃛𝑦̇²−3𝑥̈𝑦𝑦̇̈+3𝑥̇𝑦̈²𝑦̇⁵, (4) from which we determine the third-order system. Here, the invariants 𝑤₅ and 𝑤₆ are of order three, that is, equal to the order of the considered system of ODEs. By proceeding in this way for regular systems of two 𝑘th-order ODEs admitting three-dimensional Lie algebras, the number of order 𝑘 invariants is two.

To study differential invariants further, we consider the following example. Consider the system 𝑥̈𝑥̇𝑦̇=𝑓𝑒^𝑦𝑦,̇𝑥̇𝑦̇^𝑎−1,𝑒^−𝑦𝑥̈𝑦̇^𝑎−1𝑦̇²+𝑦̈+(𝑎−1)𝑒^−𝑦𝑥̇𝑦̇^𝑎−2𝑦̈𝑦̇²+𝑦̈=𝑔𝑒^𝑦𝑦,̇𝑥̇𝑦̇^𝑎−1, (5) admitting the three-dimensional Lie algebra A^𝑎,2_3,6,|𝑎|≤1,𝑎≠0,1. We arrive at the following set of invariants 𝑢=𝑒^𝑦𝑦,̇𝑣=𝑥̇𝑦̇^𝑎−1𝑥̈,𝑤=𝑥̇𝑦̇,𝑑𝑣=𝑒𝑑𝑢^−𝑦𝑥̈𝑦̇^𝑎−1𝑦̇²+𝑦̈+(𝑎−1)𝑒^−𝑦𝑥̇𝑦̇^𝑎−2𝑦̈𝑦̇²+𝑦̈. (6) This set involves three invariants 𝑢,𝑣, and 𝑤 which play the role of basis of invariants for the considered system of second-order ODEs admitting the three-dimensional Lie algebra 𝐴^𝑎,2_3,6,|𝑎|≤1,𝑎≠0,1. This set is a complete set of invariants for the system. In the complete set, one of the invariants is obtained by differentiation. We state a result on invariant differentiation of invariants corresponding to a Lie algebra admitted by any system of ODEs which is already well-known for scalar ODEs in the literature.

Proposition 1. If 𝑢 and 𝑣 are invariants of a Lie algebra admitted by any system of ODEs, then, 𝑑𝑣/𝑑𝑢 is also its invariant.

The proof is identical (except here we have many dependent variables) to the scalar case and uses the operator identity 𝑋𝐷_𝑡−𝐷_𝑡𝑋=−(𝐷_𝑡𝜉)𝐷_𝑡 (see, e.g., [17]).

One can write 𝑑𝑣=𝐷𝑑𝑢^𝑡𝑣𝐷_𝑡𝑢=𝐷𝑣, (7) where the invariant differentiation operator is 𝐷=𝐷_𝑡𝑢⁻¹𝐷_𝑡. (8) There arises a natural question. Can we construct a 𝑘th-order system from a basis of invariants corresponding to the symmetry vectors of a second-order system? We answer this question.

Suppose that we have a complete set of invariants representing a second-order system of ODEs admitting a three-dimensional Lie algebra, say for the above case 𝐴^𝑎,2_3,6,|𝑎|≤1,𝑎≠0,1. Now if we differentiate both of the highest-order invariants from this set with respect to another basic invariant, we get two new invariants and the total number of invariants become six for this case, including the previous four. These two new invariants play the role of third-order invariants representing the third-order system of ODEs admitting the three-dimensional Lie algebra 𝐴^𝑎,2_3,6,|𝑎|≤1,𝑎≠0,1. We therefore have for this case 𝑢=𝑒^𝑦𝑦,̇𝑣=𝑥̇𝑦̇^𝑎−1𝑥̈,𝑤=𝑥̇𝑦̇,𝑑𝑣𝑑𝑢=𝐷𝑣,𝑑𝑤𝑑𝑑𝑢=𝐷𝑤,²𝑣𝑑𝑢²=𝐷²𝑣, (9) where 𝐷=𝑒^−𝑦(𝑦̇²+𝑦̈²)⁻¹𝐷_𝑡 and 𝐷𝑤 and 𝐷²𝑣 are third-order invariants. We can thus construct the following third-order system 𝑒^−𝑦𝑥⃛𝑥̇𝑦̇−𝑒^−𝑦𝑥̈𝑦𝑥̇̈+𝑥̇𝑦̈𝑥̇²𝑦̇²𝑦̇²+𝑦̈(),𝑒=𝑓𝑢,𝑣,𝑤,𝐷𝑣_−2𝑦𝑦̇^𝑎−3−𝑥̈𝑦̇⁵+(2𝑎−5)𝑥̈𝑦̇³𝑦̈−(𝑎−1)𝑥̇𝑦̇⁴𝑦̈+2(𝑎−1)𝑥̈𝑦𝑦̇̈²𝑦̇²+𝑦̈³⁻¹+𝑒^−2𝑦𝑦̇^𝑎−3(𝑎−1)(𝑎−5)𝑥̇𝑦̇²𝑦̈²+(𝑎−2)𝑥̇𝑦̈³+𝑦̇²𝑥⃛𝑦̇²+𝑦̈+𝑦̇²()𝑥̇̇𝑥̈𝑦⃛×𝑦̇𝑎−1𝑦−₂+𝑦̈³⁻¹=𝑔(𝑢,𝑣,𝑤,𝐷𝑣). (10) We have constructed a system of two third-order ODEs admitting the three-dimensional Lie algebra 𝐴^𝑎,2_3,6,|𝑎|≤1,𝑎≠0,1 from their complete set of invariants. This system of ODEs is represented by a complete set of invariants which contains six functionally independent invariants, namely, 𝑢,𝑣,𝑤,𝐷𝑣,𝐷𝑤, and 𝐷²𝑣, where three of them 𝑢,𝑣,and𝑤 are a basis of invariants and two of them 𝐷𝑤,𝐷²𝑣 are third-order invariants.

Now if we repeat this process on the third-order invariants, we obtain eight invariants including the previous ones which form a complete set of invariants for the fourth prolonged group. From these, we can construct a system of two fourth-order ODEs admitting the three-dimensional Lie algebra 𝐴^𝑎,2_3,6,|𝑎|≤1,𝑎≠0,1. We can generalize this to a system of two 𝑘th-order ODEs. However, there are some nonstandard cases to look at first. These behave differently. There are four cases 𝐴³_3,1,𝐴³_3,5,𝐴⁴_3,1, and 𝐴¹_3,8 in which two of them 𝐴³_3,1 and 𝐴³_3,5 yield singular invariants and the other two 𝐴⁴_3,1 and 𝐴¹_3,8 give a single equation and do not form a system.

2.1. Singular Case

In both of the cases 𝐴³_3,1 and 𝐴³_3,5, we have an extra condition, due to which the rank of the coefficient matrix on the solution manifold is less than the rank of the coefficient matrix on the generic manifold. So singular invariants are obtained in both of these cases.

Consider 𝐴³_3,1≃𝜕_𝑥,𝑡𝜕_𝑥,𝑦𝜕_𝑥. We find the following set of singular invariants 𝑢=𝑡,𝑣=𝑦,𝑤=𝑥̈,𝐷𝑣=𝑦,̇𝑦̈=0, (11) where 𝐷=𝐷_𝑡 and the corresponding system are 𝑥̈=𝑓(𝑢,𝑣,𝐷𝑣),𝑦̈=0. (12) In both of the cases, the singularity of the invariants is removed in the third- and higher-order systems. In the case of 𝐴³_3,1, we get the following complete set of invariants: ̇𝐷𝑢=𝑡,𝑣=𝑦,𝐷𝑣=𝑦,₂𝑣=𝑦,̈𝑤=𝑥⃛𝑦̈−𝑥̈𝑦,⃛𝐷³𝑣=𝑦,⃛ (13) and the corresponding third-order system is ⃛𝑥̈𝑦⃛𝑥=𝑦̈+1𝑦̈𝑓𝑢,𝑣,𝐷𝑣,𝐷²𝑣,𝑦⃛=𝑔𝑢,𝑣,𝐷𝑣,𝐷²𝑣. (14) In this case, 𝑢,𝑣,and𝑤 form a basis of invariants, and here the third-order invariants are 𝑤 and 𝐷³𝑣, where 𝐷=𝐷_𝑡. For the other case 𝐴³_3,5, it is easy to see that ̇𝐷𝑢=𝑡,𝑣=𝑦,𝐷𝑣=𝑦,₂̈𝑥⃛𝑣=𝑦,𝑤=𝑥̈,𝐷³𝑣=𝑦,⃛ (15) where again 𝐷=𝐷_𝑡. For each of the remaining cases 𝐴⁴_3,1 and 𝐴¹_3,8, we obtain the same set of second-order invariants 𝑦̇𝑢=𝑥,𝑣=𝑦,𝐷𝑣=𝑥̇,𝐷²𝑦̈𝑣=𝑥̇²−𝑦𝑥̇̈𝑥̇³, (16) for which the highest-order invariant is only one and 𝐷=𝑥̇⁻¹𝐷_𝑡. So a system of second-order ODEs cannot be constructed. Only a single scalar equation is obtained, namely ̈𝑦̇𝑦=𝑥̇𝑥̈+𝑥̇²𝑔(𝑢,𝑣,𝐷𝑣). (17) However, in both of these cases, this behavior changes for higher orders. In fact for 𝐴⁴_3,1, we have the third-order invariants ⃛𝑤=𝑥𝜙^′′(𝑥)𝑥̇⁴−3𝑥̈²𝜙^′′(𝑥)+𝑥̇²𝑥̈𝜙^′′′(𝑥)𝑥̇⁵,𝐷³𝑦⃛𝑣=𝑥̇³+3𝑥̈²−𝑥̇𝑥⃛𝑦̇−3𝑥̇𝑥̈𝑦̈𝑥̇⁵, (18) and for 𝐴¹_3,8 the third-order invariants are 𝑥⃛𝑤=𝑥̇³−3𝑥̈²2𝑥̇⁴,𝐷³𝑦⃛𝑣=𝑥̇³+3𝑥̈²−𝑥̇𝑥⃛𝑦̇−3𝑥̇𝑥̈𝑦̈𝑥̇⁵. (19) They give rise to a regular system of third-order ODE with a complete set of invariants having the same format as the singular cases similar to (13).

We find that, for a system of two second-order ODEs admitting three-dimensional Lie algebras, there are 29 classes which are listed in Table 2 together with their second-order differential invariants.

In the case of a system of two third- and higher-order ODEs admitting three-dimensional Lie algebras, there are 31 canonical forms, and these can be obtained by invariant differentiation as described above. Note here that 𝐴⁴_3,1 and 𝐴¹_3,8 are admitted by a system of two third- and higher-order ODEs whereas they do not form a system of two second-order ODEs.

We state these results in the form of a theorem.

Theorem 2. A regular system of two 𝑘th-order ODEs (𝑘≥3) admitting three-dimensional Lie algebras can be represented by a complete set of invariants which contains 2𝑘 complete functionally independent invariants in one of the following forms: (a)𝑢,𝑣,𝑤,𝑑𝑣,𝑑𝑢𝑑𝑤,𝑑𝑑𝑢²𝑣𝑑𝑢²𝑑,…,^𝑘−2𝑤𝑑𝑢^𝑘−2,𝑑^𝑘−1𝑣𝑑𝑢^𝑘−1,(b)𝑢,𝑣,𝑑𝑣,𝑑𝑑𝑢²𝑣𝑑𝑢²𝑑,𝑤,³𝑣𝑑𝑢³,𝑑𝑤,𝑑𝑑𝑢⁴𝑣𝑑𝑢⁴𝑑,…,^𝑘−3𝑤𝑑𝑢^𝑘−3,𝑑^𝑘𝑣𝑑𝑢^𝑘. (20) In both (a) and (b), 𝑢,𝑣, and 𝑤 form a basis of invariants, where the highest order invariants in (a) are 𝑑^𝑘−2𝑤/𝑑𝑢^𝑘−2,𝑑^𝑘−1𝑣/𝑑𝑢^𝑘−1 and in (b) they are 𝑑^𝑘−3𝑤/𝑑𝑢^𝑘−3,𝑑^𝑘𝑣/𝑑𝑢^𝑘.
The total number of systems of two 𝑘th-order ODEs having the above structure is thirty-one, in which form (b) corresponds to the cases 𝐴³_3,1,𝐴³_3,5,𝐴⁴_3,1, and 𝐴¹_3,8 and (a) to the remainder.

3. Integrability

There are basically two approaches to integrability using Lie point symmetries. One is that of successive reduction of order using differential invariants for scalar ODEs. The other is that of canonical forms which may apply to scalar as well as system of ODEs, although the latter is not straightforward. However, the canonical forms approach only applies when equations are classified according to the symmetry Lie algebras. In Table 2, we have given the complete set of second-order differential invariants of three-dimensional Lie algebras admitted by systems of two second-order ODEs and also classified systems of two second-order ODEs for their point symmetries together with their canonical forms. We provide two approaches for the integration procedure on the basis of these canonical forms. The first is a general approach depending on the subdivision of the canonical forms based on general observations. The second is the differential invariants approach based on the representation of the canonical forms in terms of first-order differential and algebraic invariants.

3.1. General Integration Approach

In this approach, we judiciously subdivide all the canonical forms into four types. They are as follows.

Type I: 𝐴³_3,1⋍𝜕_𝑥,𝑡𝜕_𝑥,𝑦𝜕_𝑥,𝑥̈=𝑓𝑡,𝑦,𝑦̇,𝑦̈=0. (21)

Type II: 𝐴³_3,5⋍𝜕_𝑥,𝑡𝜕_𝑥,𝑥𝜕_𝑥,𝑦̈=𝑔𝑡,𝑦,𝑦̇,𝑥̈=0. (22)

Type III: 𝐴⁴_3,2⋍𝜕_𝑥,𝑡𝜕_𝑡+𝑥𝜕_𝑥,𝑡𝜕_𝑥,𝑡̈𝑦̇𝑡𝑦̈𝑥=𝑓𝑦,𝑡,1+𝑦̇𝑦̇,𝐴=𝑔𝑦,𝑡_33,4⋍𝜕_𝑥,𝑡𝜕_𝑥,−𝜕_𝑡+𝑥𝜕_𝑥,𝑒^𝑡̈𝑦̇,𝑦̈𝑥=𝑓𝑦,𝑦̇𝑦̇,𝐴=𝑔𝑦,_43,5⋍𝜕_𝑥,𝑡𝜕_𝑥,𝑥𝜕_𝑥+𝜕_𝑦,𝑒^−𝑦̈𝑦̇,̈𝑦̇,𝐴𝑥=𝑓𝑡,𝑦=𝑔𝑡,_𝑎,33,6⋍𝜕_𝑥,𝑡𝜕_𝑥,(1−𝑎)𝑡𝜕_𝑡+𝑥𝜕_𝑥,𝑡^{(2𝑎−1)/(𝑎−1)}̈𝑦̇𝑡𝑦̈𝑥=𝑓𝑦,𝑡,1+𝑦̇𝑦̇,𝐴=𝑔𝑦,𝑡_𝑎,33,7⋍𝜕_𝑥,𝑡𝜕_𝑥,−1+𝑡²𝜕_𝑡+(𝑎−𝑡)𝑥𝜕_𝑥,𝑥̈𝑦̇^−3/2𝑒^𝑎tan𝑡=𝑓𝑦,𝑦̇1+𝑡²,2𝑡+1+𝑡²𝑦̇𝑦̈=𝑔𝑦,𝑦̇1+𝑡². (23)

Type IV: Remaining canonical forms except those given above.

We notice that the Types I and II systems comprise linear equations. The Type I system is completely integrable. Also that the Type II system is uncoupled and depends on two quadratures of the 𝑦 equation for its solution. Further, we observe that the Type III has equations that comprise scalar second-order ODEs in the dependent variable 𝑦. In view of these observations, we are led to the following definitions.

Definition 3. A system of two second-order ODEs is said to be partially uncoupled if one of its equations constitutes a scalar second-order equation in one of the dependent variables and the other has both.

It is easily seen that the Types I to III equations are partially uncoupled.

Definition 4. A partially uncoupled system of two second-order ODEs is partially linear if one of its equations forms a scalar linear second-order equation in its own right.

We see that the Types I and II systems are partially linear.

Definition 5. A system of two second-order ODEs is said to be partially linearizable by point transformation if it can be transformed via an invertible transformation to a partially linear form.

The following system 𝑥̈=𝑦̇⁴,𝑦̈+3𝑦𝑦̇+𝑦³=0 (24) is partially linearizable by a point transformation as the equation in 𝑦 is the well-known Painlevé-Ince equation which is linearizable via a point transformation.

Let {𝑋_𝑖} be a basis of a three-dimensional algebra and let the rank of the coefficient matrix associated with these operators be 𝑟.

We formulate the following theorems the proofs of which are evident.

Theorem 6. A system of two second-order ODEs is partially linearizable by point transformation if and only if it admits the Lie algebra 𝐴_3,1 or 𝐴_3,5 with rank 𝑟=1 in both cases.

Theorem 7. A system of two second-order ODEs can be reduced to a partially uncoupled system by point transformation if and only if it admits one of the Lie algebras 𝐴⁴_3,2,𝐴³_3,4,𝐴⁴_3,5, 𝐴^𝑎,3_3,6, and 𝐴^𝑎,3_3,7 with rank 𝑟=2 in all cases.

The proofs follow from the above listed Types I to III algebras and representative equations.

Generally, we can integrate Type I completely. Types II and III can be integrated when the associated second-order equation in the system is integrable. Moreover, only particular cases of Type IV can be integrated depending on the given Lie algebra and corresponding system of ODEs. Some of these difficulties were also pointed out for systems of two second-order ODEs that admit four symmetries [12]. Here, we further discuss all the cases in detail before embarking on a further systematic study that captures the main results.

Type I is trivially integrable by quadratures.

The Type II system is integrable if the associated second-order ODE in 𝑦 admits a two-dimensional Lie algebra and Lie's four canonical forms for scalar second-order ODEs become applicable. So, we have Lie reducibility here.

In the case of Type III, one requires one more symmetry for the associated equation in 𝑦 as it already has translations in 𝑡 symmetry. As an example we consider the system 𝑥̈=𝑒^−𝑡𝑦³𝑦̇⁴,𝑦̈=𝑦̇²𝑦⁻¹−𝑦². (25) The system (25) has symmetry generators 𝑋₁=𝜕_𝑥,𝑋₂=𝑡𝜕_𝑥,𝑋₃=−𝜕_𝑡+𝑥𝜕_𝑥 and its Lie algebra is three-dimensional. The associated equation in 𝑦 is 𝑦̈=𝑦̇²𝑦⁻¹−𝑦². Its Lie algebra is two-dimensional spanned by 𝑌₁=𝜕_𝑡,𝑌₂=𝑡𝜕_𝑡−2𝑦𝜕_𝑦. By using the transformation 𝑡=𝑦,𝑥=𝑥,𝑦=𝑡, (26) the associated equation in transformed form becomes 𝑦̈‌=𝑡²𝑦̇‌³−𝑦̇𝑡⁻¹. We have 𝑌₁=𝜕_𝑦,𝑌₂=𝑦𝜕_𝑦−2𝑡𝜕_𝑡, which is Type III in Lie's classification of scalar second-order ODEs. Solving it in transformed form by Lie's method and reverting back to the original variables, we obtain 𝑦=2𝑐₁²sech²𝑐₁(𝑡−𝑐₂), where 𝑐₁ and 𝑐₂ are constants. By using the value of 𝑦 in system (25) and then solving for 𝑥, we deduce 𝑥=2048𝑐₁¹⁸𝑒^−𝑡sech¹⁴𝑐₁𝑡−𝑐₂×tanh⁴𝑐₁𝑡−𝑐₂𝑑𝑡𝑑𝑡+𝑐₃𝑡+𝑐₄. (27) In general, the Type III systems of second-order ODEs are integrable if the associated second-order ODEs in 𝑦 have another symmetry generator besides translations in 𝑡, so that Lie's classification of scalar second-order ODEs admitting two-dimensional algebras is applicable here as well.

Theorem 8. If a singular system of two second-order ODEs admits the solvable symmetry algebra 𝐴³_3,1, then, it is partially linearizable via point transformation and its general solution can be trivially obtained by quadratures.

The proof of this follows from the Type I canonical form.

Theorem 9. If a singular system of two second-order ODEs admits the three-dimensional solvable symmetry algebra 𝐴³_3,5, then, it can be reduced to a partially uncoupled system and its general solution can be obtained by quadratures from the general solution of the associated transformed scalar second-order ODE.

The proof here follows from the Type II canonical form.

For systems that fall in the Type IV category one requires more insight and the examples given in [12] for systems that admit four symmetries become relevant. Their integrability needs more than just the reductions via invariants. This category also has the nonsolvable algebras. We consider this in detail next.

3.2. Differential Invariant Approach

In this approach, we use the basis of invariants representing all the canonical forms which we have given in Table 2. For each regular canonical form, we find the reduced system in terms of a first-order equation and an algebraic equation based on the basis of invariants. These can easily be seen from Table 2.

We have subdivided all the reduced systems in two main categories, namely, Category A for those admitting three-dimensional solvable Lie algebras and Category B for those admitting three-dimensional nonsolvable Lie algebras. The Types I to III of the previous subsection all fall in Category A. The systems admitting nonsolvable algebras comprise Category B with the rest firmly falling in Category A. It is interesting that, in both categories, uncoupled cases arise when the choice of taking one of the second-order invariant 𝑤 is unique.

We present the integration strategy for Category A for regular systems. The systems of Category B are not amenable to such integration strategies. This is quite straightforward to work out if one commences with the simplest system in this category. We further subdivide Category A into coupled and uncoupled systems. All partially linear cases fall in the uncoupled systems. Here, we give details of integration for each case of Category A, which depends on the assumption that we can solve 𝑑𝑣/𝑑𝑢=ℎ in both the coupled and the uncoupled systems. At the end of this section, we present some physical examples with detailed calculations.

Note here that, in both the uncoupled and coupled cases, for ease and simplicity of the calculations, we only mention the relevant algebra of the considered system and variables 𝑢,𝑣,and𝑤 as well as 𝑑𝑣/𝑑𝑢 in precise form. The details are given in Table 2. In the following, we assume that 𝑣=𝜆(𝑢,𝑐₁) is the explicit solution of each of the first order ODEs 𝑑𝑣/𝑑𝑢=ℎ(𝑢,𝑣) listed in Table 2. The argument presented below is similar to that used in the discussion of reduction to quadrature for the scalar ODEs (see [1–3]). Moreover, 𝑓,𝑔,ℎ,𝑘,𝜙,𝜓, and 𝜆 are taken as arbitrary functions and 𝑐₁,𝑐₂,and𝑐₃ as well as 𝑐₄ are arbitrary integration constants.

3.2.1. Uncoupled Cases

𝐴⁴_3,2. Let 𝑣=𝜆(𝑢,𝑐₁). Then by using the values of 𝑢 and 𝑣 in Table 2, we have 𝑡𝑦̇=𝜆𝑦,𝑐₁. (28) Thus, we can obtain a further quadrature in 𝑦. Therefore, the equation 𝑡𝑥̈=𝑓(𝑦,𝑡𝑦)̇ is solvable by two quadratures upon insertion of the 𝑦 solution.

𝐴¹_3,3. Let 𝑣=𝜆(𝑢,𝑐₁). Then by invoking the values of 𝑢 and 𝑣 as listed in Table 2, we obtain 𝑥̇=𝑦𝜆̇𝑦,𝑐₁. (29) After some manipulations with 𝑤=𝑓(𝑢,𝑣), we find 𝑥̇=−1∫𝜆𝑦,𝑐₁𝑓𝑦,𝜆𝑦,𝑐₁𝑑𝑦+𝑐₂. (30) By using (30) in (29), we determine 𝑦𝜆̇𝑦,𝑐₁𝜆𝑦,𝑐₁𝑓𝑦,𝜆𝑦,𝑐₁𝑑𝑦+𝑐₂=−1, (31) which implies that 𝑦=𝜙𝑡,𝑐₁,𝑐₂,𝑐₃. (32) Utilizing (32) in (30), after some further calculations, we deduce 𝑥=𝜓𝑡,𝑐₁,𝑐₂,𝑐₃,𝑐₄. (33) Hence, (32) and (33) are the solutions of the system of ODEs admitting 𝐴¹_3,3.

𝐴²_3,3. Let 𝑣=𝜆(𝑢,𝑐₁). By using the values of 𝑢 and 𝑣 as listed in Table 2, we arrive at𝑥̇=±2𝑦̇+2𝑦̇²𝜆𝑦,𝑐₁. (34) After calculations with 𝑤=𝑓(𝑢,𝑣), we find 𝑥̇=−𝑦̇𝑓𝑦,𝜆𝑦,𝑐₁𝑑𝑦+𝑐₂. (35) Invoking (35) in (34) and solving for 𝑦̇, we obtain ̇2𝑦=∫𝑓(𝑦,𝜆(𝑦,𝑐₁))𝑑𝑦+𝑐₂²−2𝜆𝑦,𝑐₁, (36) which implies that 𝑦=𝜙𝑡,𝑐₁,𝑐₂,𝑐₃. (37) After some computations and using (37) in (35), we determine 𝑥=𝜓𝑡,𝑐₁,𝑐₂,𝑐₃,𝑐₄. (38) It is hence the case that (37) and (38) are the solutions of the system of ODEs corresponding to 𝐴²_3,3.

𝐴³_3,4. Let 𝑣=𝜆(𝑢,𝑐₁). Using the values of 𝑢 and 𝑣, we obtain 𝑦̇=𝜆𝑦,𝑐₁, (39) which implies that 𝑦=𝜙𝑡,𝑐₁,𝑐₂. (40) Now utilizing (39) and (40) in 𝑤=𝑓(𝑢,𝑣), we deduce that 𝑥=𝑒^−𝑡𝑓𝜙,𝜙_𝑡𝑑𝑡𝑑𝑡+𝑐₃𝑡+𝑐₄, (41) which results in 𝑥=𝜓𝑡,𝑐₁,𝑐₂,𝑐₃,𝑐₄. (42) Now (40) and (42) are the solutions of the system of ODEs possessing 𝐴³_3,4.

𝐴¹_3,5. Let 𝑣=𝜆(𝑢,𝑐₁). Utilizing the values of 𝑢 and 𝑣 as listed, we obtain 𝑥̇=𝜆𝑦,𝑐₁. (43) After some manipulations with 𝑤=𝑓(𝑢,𝑣), we find 𝑑𝑦𝑐₂exp∫𝑓𝑦,𝜆𝑢,𝑐₁𝑑𝑦=𝑡+𝑐₃. (44) Inverting, we have 𝑦=𝜙𝑡,𝑐₁,𝑐₂,𝑐₃. (45) By using (45) in (43), we have that 𝑥=𝜓𝑡,𝑐₁,𝑐₂,𝑐₃,𝑐₄. (46) Thus, (45) and (46) are the solutions of the system of ODEs corresponding to 𝐴¹_3,5.

For 𝐴⁴_3,5,𝐴^𝑎,3_3,6, and 𝐴^𝑎,3_3,7, the integration procedure works the same as in the case 𝐴³_3,4.

3.2.2. Coupled Cases

𝐴¹_3,1. Let 𝑣=𝜆(𝑢,𝑐₁). Using the values of 𝑢 and 𝑣 as listed in Table 2, we obtain 𝑦̇=𝜆𝑥̇,𝑐₁. (47) After calculations with 𝑤=𝑓(𝑢,𝑣), we arrive at 𝑥=𝜙𝑡,𝑐₁,𝑐₂,𝑐₃. (48) By utilizing (48) in (47), in the same manner as in the previous cases, we find that 𝑦=𝜓𝑡,𝑐₁,𝑐₂,𝑐₃,𝑐₄. (49) Hence (48) and (49) are the solutions of the system of ODEs corresponding to 𝐴¹_3,1.

𝐴²_3,1. The procedure of integration is the same as in the case 𝐴¹_3,1.

𝐴¹_3,2. Let 𝑣=𝜆(𝑢,𝑐₁). Using the values of 𝑢 and 𝑣 as listed in Table 2, we determine 𝑥̇=𝑦𝜆̇𝑦𝑒̇^𝑦,𝑐₁. (50) After some manipulations with 𝑤=𝑓(𝑢,𝑣), we find 𝑥̇=exp𝜆𝑦𝑒̇^𝑦,𝑐₁+𝑦𝑒̇^𝑦𝑔𝑦𝑒̇^𝑦,𝜆𝑦𝑒̇^𝑦,𝑐₁×𝑓𝑦𝑒̇^𝑦,𝜆𝑦𝑒̇^𝑦,𝑐₁𝑑𝑦𝑒̇^𝑦×𝑦𝑒̇^𝑦1+𝜆𝑦𝑒̇^𝑦,𝑐₁𝑓𝑦𝑒̇^𝑦,𝜆𝑦𝑒̇^𝑦,𝑐₁⁻¹. (51) From the second equation of the system in Table 2 and some manipulations with (50), we find that 𝑑𝑦𝑒̇^𝑦=𝑦̇𝑑𝑡₂𝑒^𝑦𝜆𝑦𝑒̇^𝑦,𝑐₁1+𝜆𝑦𝑒̇^𝑦,𝑐₁𝑓𝑦𝑒̇^𝑦,𝜆𝑦𝑒̇^𝑦,𝑐₁𝜆𝑦𝑒̇^𝑦,𝑐₁+𝑦𝑒̇^𝑦𝑔𝑦𝑒̇^𝑦,𝜆𝑦𝑒̇^𝑦,𝑐₁. (52) Some further computations give 𝑦𝑒̇^𝑦=𝜙𝑦+𝑐₂, (53) which implies that 𝑦=ℎ𝑡,𝑐₁,𝑐₂,𝑐₃. (54) Using (53) and (54) in (51), we find 𝑥=𝑘𝑡,𝑐₁,𝑐₂,𝑐₃,𝑐₄. (55) Therefore, (54) and (55) are the solutions of the system of ODEs corresponding to 𝐴¹_3,2.

𝐴²_3,2. The procedure of integration is the same as the case 𝐴¹_3,5.

𝐴³_3,2. The procedure of integration is similar to the case 𝐴¹_3,3.

𝐴³_3,3. Let 𝑣=𝜆(𝑢,𝑐₁). Using the values of 𝑢 and 𝑣 as listed in Table 2, we obtain 𝑥̇𝑦̇1=𝜆𝑦−𝑥̇,𝑐₁. (56) Using the chain rule and some manipulations with 𝑤=𝑓(𝑢,𝑣), we arrive at 1𝑦=𝜙𝑦−𝑥̇,𝑐₁,𝑐₂. (57) Further simplifications give ̇1𝑥=𝑦−𝜙⁻¹𝑦,𝑐₁,𝑐₂. (58) Utilizing (57) and (58) in (56), we find that 𝑦̇𝑦−𝜙⁻¹𝑦,𝑐₁,𝑐₂𝜆𝜙⁻¹𝑦,𝑐₁,𝑐₂,𝑐₁=1, (59) which implies that 𝑦=ℎ𝑡,𝑐₁,𝑐₂,𝑐₃. (60) Invoking (60) in (58), we have 𝑥=𝑘𝑡,𝑐₁,𝑐₂,𝑐₃,𝑐₄. (61) Thus, (60) and (61) are the solutions of the system of ODEs possessing 𝐴³_3,3.

𝐴¹_3,4. The procedure of integration is similar to the case 𝐴¹_3,3.

𝐴²_3,4. The procedure of integration is the same as the case 𝐴³_3,3.

𝐴²_3,5. Let 𝑣=𝜆(𝑢,𝑐₁). Utilizing the values of 𝑢 and 𝑣 as listed in Table 2, we obtain 𝑦𝑒̇^𝑦=𝜆𝑥̇,𝑐₁. (62) By the chain rule and some simplifications with 𝑤=𝑓(𝑢,𝑣), we find 𝑥̇=𝜙𝑦,𝑐₁,𝑐₂. (63) Substituting (63) in (62) and further manipulations give 𝑦=𝜓𝑥,𝑐₁,𝑐₂,𝑐₃. (64) Invoking this in (63), we deduce that 𝑥=ℎ𝑡,𝑐₁,𝑐₂,𝑐₃,𝑐₄. (65) Now after substitution of (65), (64) takes the form𝑦=𝜓𝑡,𝑐₁,𝑐₂,𝑐₃,𝑐₄. (66) Here, (65) and (66) are the solutions of the system of ODEs corresponding to 𝐴²_3,5.

𝐴^𝑎,1_3,6. The procedure of integration is the same as the case 𝐴¹_3,3.

𝐴^𝑎,2_3,6. The procedure of integration is again similar to the case 𝐴¹_3,2.

𝐴^𝑎,1_3,7. The procedure of integration corresponds to the case 𝐴¹_3,3 and some details are given in the case 𝐴^𝑎,2_3,7.

𝐴^𝑎,2_3,7. Let 𝑣=𝜆(𝑢,𝑐₁). Using the values of 𝑢 and 𝑣 as listed in Table 2, we obtain 𝑦𝑒̇^{̇𝑥−𝑎arctan}√1+𝑥̇²=𝜆𝑦+arctan𝑥̇,𝑐₁. (67) After manipulations with 𝑤=𝑓(𝑢,𝑣), we find ̇𝑓𝑥=tan𝑢,𝜆𝑢,𝑐₁𝑑𝑢𝜆𝑢,𝑐₁+𝑓𝑢,𝜆𝑢,𝑐₁+𝑐₂. (68) Invoking (68) in (67), we determine 𝑦̇=𝜆𝑢,𝑐₁𝑓sec𝑎𝑢,𝜆𝑢,𝑐₁𝑑𝑢𝜆𝑢,𝑐₁+𝑓𝑢,𝜆𝑢,𝑐₁+𝑐₂𝑎𝑓×exp𝑢,𝜆𝑢,𝑐₁𝑑𝑢𝜆𝑢,𝑐₁+𝑓𝑢,𝜆𝑢,𝑐₁+𝑎𝑐₂. (69) We have 𝑑𝑢=𝑦̇𝜆𝑑𝑡𝑢,𝑐₁+𝑓𝑢,𝜆𝑢,𝑐₁𝜆𝑢,𝑐₁. (70) In (68)–(70), 𝑢=𝑦+arctan𝑥̇. (71) Also, we find that 𝑢=𝜙⁻¹𝑡+𝑐₃,𝑦+arctan𝑥̇=𝜙⁻¹𝑡+𝑐₃. (72) Using (69) and (72) in (70), we deduce 𝑢=𝜓𝑡,𝑐₁,𝑐₂,𝑐₃, (73) which shows that 𝑦=ℎ𝑡,𝑐₁,𝑐₂,𝑐₃. (74) By invoking (74) in (68), we have that 𝑥=𝑘𝑡,𝑐₁,𝑐₂,𝑐₃,𝑐₄. (75) Thus, (74) and (75) are the solutions of the system of ODEs corresponding to 𝐴^𝑎,2_3,7.

In the view of the preceding reductions, we have the following theorem.

Theorem 10. If a regular system of two second-order ODEs admits a three-dimensional solvable symmetry algebra, then, the general solution of the regular system can be obtained by quadratures from the general solution of the invariant representation of the regular system given by the first-order ordinary differential equation and algebraic equation.

Theorem 11. If a system of two 𝑘th-order (𝑘≥3) ODEs possess a three-dimensional solvable symmetry algebra, then, the general solution of the system can be deduced by quadratures from the general solution of the invariant representation given by one of the following systems: 𝑑(a)^𝑘−1𝑣𝑑𝑢^𝑘−1=𝑓𝑢,𝑣,𝑤,𝑑𝑣,𝑑𝑢𝑑𝑤,𝑑𝑑𝑢²𝑣𝑑𝑢²𝑑,…,^𝑘−3𝑤𝑑𝑢^𝑘−3,𝑑^𝑘−2𝑣𝑑𝑢^𝑘−2,𝑑^𝑘−2𝑤𝑑𝑢^𝑘−2=𝑔𝑢,𝑣,𝑤,𝑑𝑣,𝑑𝑢𝑑𝑤,𝑑𝑑𝑢²𝑣𝑑𝑢²𝑑,…,^𝑘−3𝑤𝑑𝑢^𝑘−3,𝑑^𝑘−2𝑣𝑑𝑢^𝑘−2,𝑑(b)^𝑘𝑣𝑑𝑢^𝑘=𝑓𝑢,𝑣,𝑑𝑣,𝑑𝑑𝑢²𝑣𝑑𝑢²𝑑,𝑤,³𝑣𝑑𝑢³,𝑑𝑤,𝑑𝑑𝑢⁴𝑣d𝑢⁴𝑑,…,^𝑘−4𝑤𝑑𝑢^𝑘−4,𝑑^𝑘−1𝑣𝑑𝑢^𝑘−1,𝑑^𝑘−3𝑤𝑑𝑢^𝑘−3=𝑔𝑢,𝑣,𝑑𝑣,𝑑𝑑𝑢²𝑣𝑑𝑢²𝑑,𝑤,³𝑣𝑑𝑢³,𝑑𝑤,𝑑𝑑𝑢⁴𝑣𝑑𝑢⁴𝑑,…,^𝑘−4𝑤𝑑𝑢^𝑘−4,𝑑^𝑘−1𝑣𝑑𝑢^𝑘−1. (76)

The proof follows from Theorem 2 and the discussion preceding Theorem 10.

3.2.3. Illustrative Examples

Here, we present some familiar physical examples to illustrate the above general integration procedure which we have given in detail for each of the case of system of ODEs admitting solvable three-dimensional Lie algebras. We consider two different cases, namely, one from the uncoupled cases and one from the coupled cases. We also present a case of nonsolvable Lie algebras. Here, we observe interesting results rather than integration.

(i) Consider the system (25) once again from a new angle. We have 𝑥̈=𝑒^−𝑡𝑦³𝑦̇⁴,𝑦̈=𝑦̇²𝑦⁻¹−𝑦². (77) The system (25) has symmetry generators 𝑋₁=𝜕_𝑥,𝑋₂=𝑡𝜕_𝑥,𝑋₃=−𝜕_𝑡+𝑥𝜕_𝑥 and its Lie algebra is three-dimensional which is 𝐴³_3,4 in our classification. From Table 2, the first-order differential invariants for this case are 𝑢=𝑦,𝑣=𝑦,̇ (78) and the reduced system of first-order equation and algebraic equation is 𝑑𝑣()().𝑑𝑢=𝑔𝑢,𝑣,𝑤=𝑓𝑢,𝑣 (79) The solution of 𝑑𝑣/𝑑𝑢 in system (79), after substituting the value of 𝑔 from (25), gives 𝑣=±𝑢2𝑐^∗₁−𝑢, (80) where 𝑐^∗₁ is constant. Reverting back to the original variables and after solving for 𝑦, we obtain 𝑦=2𝑐₁²sech²𝑐₁𝑡−𝑐₂, (81) where 𝑐₁=𝑐^∗₁/2 and 𝑐₂ are constants. Substituting the values in 𝑤 and solving for 𝑥, we find 𝑥=2048𝑐₁¹⁸𝑒^−𝑡sech¹⁴𝑐₁𝑡−𝑐₂×tanh⁴𝑐₁𝑡−𝑐₂𝑑𝑡𝑑𝑡+𝑐₃𝑡+𝑐₄. (82) Both (81) and (82) provide the solution of system (25).

(ii) The classical Kepler problem in reduced Cartesian coordinates is 𝑟̈⃗+𝜇𝑟⃗𝑟³=0, (83) where 𝜇 is a constant. In polar coordinates, they are 𝑟̈−𝑟𝜃̇²+𝜇𝑟⁻²=0,𝑟𝜃̈+2𝑟𝜃̇̇=0. (84) It has the three-dimensional Lie algebra of symmetries [18] spanned by 𝑋₁=𝜕_𝑡,𝑋₂=𝜕_𝜃,𝑋₃=𝑡𝜕_𝑡−23𝑟𝜕_𝑟. (85) By using the transformation 𝑡=𝑡,𝑥=𝜃,3𝑦=2ln𝑟 (86) the system (84) transforms into 𝑦̈‌+23𝑦̇²−32𝑥̇²+32𝜇𝑒^̇−2𝑦=0,̈4𝑥+3𝑥̇𝑦̇=0, (87) with associated transformed Lie algebra of symmetries 𝑋₁=𝜕_𝑡,𝑋₂=𝜕_𝑥,𝑋₃=𝑡𝜕_𝑡+𝜕_𝑦, (88) which is 𝐴¹_3,2 in our classification. From Table 2, the first-order differential invariants for this case are 𝑢=𝑦̇𝑒^𝑦̇,𝑣=𝑥𝑦̇‌, (89) and the reduced system of first-order equation and algebraic equation is 𝑑𝑣()().𝑑𝑢=𝑔𝑢,𝑣,𝑤=𝑓𝑢,𝑣 (90) By utilizing the values of 𝑓 and 𝑔 from (87), we have that 𝑑𝑣=𝑑𝑢−4𝑣+9𝜇𝑣/𝑢²−9𝑣³2𝑢−9𝜇/𝑢+9𝑣²𝑢4,𝑤=−.3𝑣 (91) Now the solution of 𝑑𝑣/𝑑𝑢 in the system (91) is 𝑢⁴𝑣²−92𝜇𝑢²𝑣²+94𝑢⁴𝑣⁴=ℎ. (92) In 𝑥 and 𝑦 variables, after some simplification, (92) and 𝑤 from (91) give 𝑑𝑒^(2/3)𝑦𝑑𝑡=±4ℎ9𝑘²+2𝜇𝑒^(2/3)𝑦−𝑒^(4/3)𝑦,𝑒^(2/3)𝑦2𝑥̇‌=𝑘, (93) where ℎ and 𝑘 are constants. Reverting back to the original variables, we deduce 𝑑𝑟𝑑𝑡=±4ℎ9𝑘²+2𝜇𝑟−1𝑟²,𝑟²𝜃̇=𝑘. (94)(95) The term on the left-hand side of (95) is the angular momentum and here it is constant from the calculations and denoted by 𝐿/𝑚. Performing a simple integration procedure, we arrive at the equation of the orbit, namely, 𝑑𝜃=𝑑𝑠∓ℎ^∗+2𝑚²𝜇𝑠/𝐿²−𝑠², (96) where 𝑠(𝜃)=1/𝑟(𝜃) and ℎ^∗=4ℎ/9𝑘⁴. This corresponds to that presented in [19].

(iii) Consider a case of nonsolvable Lie algebras, namely, the generalized Ermakov system in polar coordinates of the form 𝑟̈−𝑟𝜃̇²=1𝑟³𝑓𝜃,𝑟²𝜃̇,𝑟̈𝑟̇̇1𝜃+2𝜃=𝑟³𝑔𝜃,𝑟²𝜃̇, (97) which was considered in [20]. This system has the three symmetries 𝑋₁=𝜕_𝑡,𝑋₂=2𝑡𝜕_𝑡+𝑟𝜕_𝑟,𝑋₃=𝑡²𝜕_𝑡+𝑡𝑟𝜕_𝑟, (98) and forms the Lie algebra sl(2,𝑅) which is 𝐴⁴_3,8 in our classification. The first-order differential invariants for this case are 𝑢=𝑦,𝑣=(𝑡𝑥̇−𝑥)/𝑦̇, and the reduced system of first-order equation and an algebraic equation is 𝑑𝑣/𝑑𝑢=𝑔(𝑢,𝑣),𝑤=𝑓(𝑢,𝑣). By using these and considering the angular momentum as constant, 𝐿=𝑚𝑟²𝜃̇, we find that 𝑑𝑣/𝑑𝑢=−𝑣ℎ(𝑢,𝐿/𝑚),where𝑔(𝑢,𝑣)=−𝑣ℎ(𝑢,𝐿/𝑚). In the original variables, it is 1𝑀=2𝑟²𝜃̇²−ℎ(𝜃)𝑑𝜃, (99) where ℎ(𝜃)=ℎ(𝑢,𝐿/𝑚). Here 𝑀 is the Ermakov invariant. The classical integrability was discussed in [20].

4. Conclusion

The integrability of systems of ordinary differential equations (ODEs) which admit symmetry algebras has been of interest in the literature. The traditional symmetry reductions do not work for systems of ODEs as in the case of scalar ODEs (see, e.g., [5, 12, 17]). Here, we have focused on systems of two 𝑘th-order (𝑘≥2) ODEs that admit three-dimensional Lie algebras.

By utilizing the realizations given recently, we have provided a systematic approach for finding a complete set of differential invariants including basis of invariants corresponding to vector fields in three variables for systems of two 𝑘th-order (𝑘≥2) ODEs which admit three-dimensional Lie algebras. In particular, we presented a complete list of second-order differential invariants for systems of two second-order ODEs that admit three-dimensional Lie algebras. We have shown how invariant differentiations provide higher-order differential invariants. We have given a complete list of invariant characterizations of canonical forms for systems of two second-order ODEs that admit real Lie algebras of dimension three. These are new in the literature. In addition, we have given a procedure for the construction of two 𝑘th-order ODEs from their associated complete set of invariants. We have shown that there are 29 classes for the case of 𝑘=2 and 31 classes for the case of 𝑘≥3 of such ODEs. Furthermore, we have given a discussion of those cases in which regular systems of two second-order ODEs are not obtained for three-dimensional Lie algebras. We also have given a brief discussion on the singularity for canonical forms for a system of two second-order ODEs admitting-three dimensional Lie algebras and found two cases of singular invariants, namely, those involving the algebras 𝐴³_3,1 and 𝐴³_3,5.

We have provided an integration procedure for the canonical forms for a system of two second-order ODEs which possess three-dimensional Lie algebras. This procedure is composed of two approaches, namely, one which depends on general observations of canonical forms; the other depends on basis of differential invariants and the solution procedure is applicable only for those cases which admit three-dimensional solvable Lie algebras. The latter was extended in a natural way to 𝑘th-order (𝑘≥3) systems that admit three-dimensional Lie algebras. Specifically in the second approach, we obtain the reducibility for canonical forms for systems of two second-order ODEs which have three-dimensional Lie algebras in terms of basis of invariants; these are given in Table 2. Then, we gave details of integration for each case of solvable Lie algebras in a general manner.

We also have obtained two cases of partial linearization, namely, Types I and II. It was noted that the Type I system is trivially integrable. We have further shown how the associated scalar second-order ODEs in the Types II to III can be solved in terms of quadratures resorting to the classical Lie table for integrability of scalar ODEs. This was illustrated by means of an example. There are certain Type IV systems which have complications in their integrability in that they admit nonsolvable algebras.

We have presented familiar physical systems taken from the literature, namely, the classical Kepler problem and the generalized Ermakov systems that give rise to closed trajectories.

It would be of interest to further pursue integrability properties of Type IV systems having nonsolvable Lie algebras. Also worthy of investigation are systems that admit higher number of symmetries such as those studied in [12, 16]. We return to these in the future.

Remarks

(i)In the tables, 𝐴 is used as a place holder for the relevant real Lie algebras 𝐴^{𝑎,𝑏,𝑛}_𝑖,𝑗…(the 𝑗th algebra of dimension 𝑖 with superscripts 𝑎,𝑏, if any, indicates parameters on which the Lie algebra depends; also the column 𝑁 in the tables gives the algebra realizations; the realization is referred to by a superscript 𝑛,e.g.,𝐴¹_3,1). Further we use the notation 𝜕_𝑡=𝜕/𝜕𝑡,𝜕_𝑥=𝜕/𝜕𝑥,𝜕_𝑦=𝜕/𝜕𝑦. Finally, the elements of a basis of a given Lie algebra are named 𝑋_𝑖, where 𝑖 is less than or equal to the dimension of the underlying real Lie algebra. Here, 𝑓,𝑔,ℎ,𝑘, and 𝜙 are arbitrary functions.(ii)For the analysis of the invariant representations, it is observed that for each of the cases 𝐴⁴_3,1 and 𝐴¹_3,8, with the same set of invariants, a single equation is determined. We find the differential invariants in both cases. However, the canonical form is not obtained. Consider, for example, 𝐴⁴_3,1≃𝜕_𝑡,𝑥𝜕_𝑡,𝜙(𝑥)𝜕_𝑡,𝜙^′′(𝑥)≠0. For this, we find the following invariants 𝑢=𝑥,𝑣=𝑦,𝑑𝑣/𝑑𝑢=𝑦/̇𝑥̇,𝑑²𝑣/𝑑𝑢²=−𝑥̈𝑦/̇𝑥̇³+𝑦/̈𝑥̇² and the corresponding single equation, namely, −𝑥̈𝑦/̇𝑥̇³+𝑦/̈𝑥̇²=𝑔(𝑢,𝑣,𝑑𝑣/𝑑𝑢). For each of the cases 𝐴³_3,1 and 𝐴³_3,5, due to the extra condition, the rank of the coefficient matrix on the solution manifold is less than the rank of generic coefficient matrix. So only a set of singular invariants are obtained for these cases. Consider, for example, 𝐴³_3,1≃𝜕_𝑥,𝑡𝜕_𝑥,𝑦𝜕_𝑥. For this, we find the following set of singular invariants 𝑢=𝑡,𝑣=𝑦,𝑑𝑣/𝑑𝑢=𝑦,̇𝑤=𝑥̈,and𝑑𝑣/𝑑𝑢=𝑦̈=0 (see Table 2). The two cases 𝐴⁴_3,1 and 𝐴¹_3,8 are not given in the invariant representations listed in Table 2.(iii)The algebras 𝐴³_3,1and𝐴³_3,5 which are referred to by * give rise to singular invariants when associated with systems of two second-order ODEs as given in Table 2. We do however arrive at regular invariants associated with systems of two third-order ODEs for these cases.(iv)The algebras 𝐴⁴_3,1and𝐴¹_3,8 do not correspond to systems of two second-order ODEs and so are not presented in Table 2.

Acknowledgments

M. Ayub and M. Khan are grateful to the Higher Education Commission (HEC) of Pakistan for financial support (via Indigenous Scholarship and IRSIP program). Also F. M. Mahomed thanks the HEC for granting him a visiting professorship and the Department of Mathematics at the Quaid-i-Azam University, Islamabad, for its hospitality during the time this work was initiated. M. Ayub also thanks the Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, South Africa, for its hospitality during the time this project was completed. Finally F. M. Mahomed is grateful to the NRF of South Africa for research funding. Parts of this work was presented at the SDEA2012 Jan Conference.