Symmetry Classification of First Integrals for Scalar Linearizable Second-Order ODEs

Abstract

Symmetries of the fundamental first integrals for scalar second-order ordinary differential equations (ODEs) which are linear or linearizable by point transformations have already been obtained. Firstly we show how one can determine the relationship between the symmetries and the first integrals of linear or linearizable scalar ODEs of order two. Secondly, a complete classification of point symmetries of first integrals of such linear ODEs is studied. As a consequence, we provide a counting theorem for the point symmetries of first integrals of scalar linearizable second-order ODEs. We show that there exists the 0-, 1-, 2-, or 3-point symmetry cases. It is shown that the maximal algebra case is unique.

1. Introduction

First integrals or constants of the motion of ordinary differential equations (ODEs) are quite an active and interesting area of research at the present time. Whenever one is dealing with differential equations and especially with their solutions, you have to deal with first integrals. In fact, first integrals are the first primary steps towards finding the solutions of differential equations. First integrals have great importance in mechanics as it deals with second-order systems of equations and constants of the motion. The maximum number of symmetries for scalar 𝑛th-order ODEs was investigated by Lie [1] (see Mahomed [2]). Lie showed that scalar first-order ODEs have infinite number of point symmetries. In the case of scalar second-order ODEs Lie proved that the maximum is eight, and this is achieved by the free particle and indeed linearizable by point transformation equations. In recent work by Mahomed and Leach [3], they discovered the symmetries of the maximal cases of scalar linear 𝑛th-order ODEs, 𝑛≥3. These cases are 𝑛+1, 𝑛+2, and 𝑛+4. There is yet another contribution by Leach and Mahomed [4], in which they have found that the Lie algebra of the fundamental first integrals and their quotient of scalar linear second-order ODEs are three-dimensional and have very interesting properties. This also applies to linearizable by invertible transformations second-order ODEs which are given as examples in their paper. In the case of scalar third-order linear ODEs, Govinder and Leach [5] provided the algebraic structure of the basic first integrals. They showed that in the three equivalence classes each has certain first integrals with a specific number of point symmetries. So far, none of these authors consider the classification of the symmetries of first integrals of scalar linear 𝑛th-order ODEs, 𝑛≥1, nor even investigate what could be the maximal numbers of symmetries for the first integrals of these linear or linearizable equations. They do give insights into the algebraic structure of the fundamental first integrals and in some cases their quotients.

In this paper we give the complete classification of point symmetries for the first integrals of scalar linear second-order ODEs and the relationship between the symmetries and first integrals. For this purpose we use the projective transformations to find the different cases of symmetries for the first integrals of scalar second-order ODEs which are linear or linearizable by point transformations. Since all scalar second-order ODEs which are linear or linearizable by point transformations are transformable to the free particle equation, we utilize this as our base ODE. We find that there are the no-symmetry, one-symmetry, two-symmetry, and unique three-symmetry cases.

It is well known that the second-order ODE 𝐸𝑥,𝑦,𝑦^′,𝑦^′′=0 (1.1) is invariant under the infinitesimal generator 𝜕𝑋=𝜉𝑥,𝑦𝜕𝜕𝑥+𝜂𝑥,𝑦𝜕𝑦 (1.2) if and only if 𝑋^[2]𝐸|_𝐸=0=0, (1.3) where 𝑋^[2]=𝑋+𝜁₁𝜕𝜕𝑦^′+𝜁₂𝜕𝜕𝑦^′′, (1.4) with 𝜁₁=𝐷_𝑥𝜂−𝑦^′𝐷_𝑥(𝜉),𝜁₂=𝐷_𝑥𝜁₁−𝑦^′′𝐷_𝑥(𝜉) (1.5) in which 𝐷_𝑥 is the total differentiation operator, is called the second prolongation of the generator 𝑋.

Now we can say that (1.2) is the point symmetry of (1.1), whereas, in the case of first integrals, the first integral 𝐼=𝑓𝑥,𝑦,𝑦^′, (1.6) of the ODE (1.1), is annihilated by 𝑋, that is, (1.2) is the symmetry generator of (1.6) if and only if 𝑋^[1]𝐼=0. (1.7) Here 𝑋 annihilates 𝐼 and does not leave it invariant as in the case of symmetries of equations. Note that the procedure for finding symmetries of ODEs is different to that of finding symmetries of first integrals. In fact the symmetries of the first integrals are a subalgebra of the symmetries of the equation itself (see Kara and Mahomed [6]).

It is essential to point out that (1.1) is linearizable by point transformation to the free particle equation if and only if it is cubic in the first derivatives as 𝑦^′′=𝐴𝑥,𝑦𝑦^′3+𝐵𝑥,𝑦𝑦^′2+𝐶𝑥,𝑦𝑦^′+𝐷𝑥,𝑦, (1.8) where the functions from 𝐴 to 𝐷 satisfy the invariant conditions (see Tresse [7] and also [8]) 3𝐴_𝑥𝑥+3𝐴_𝑥𝐶−3𝐴_𝑦𝐷+3𝐴𝐶_𝑥+𝐶_𝑦𝑦−6𝐴𝐷_𝑦+𝐵𝐶_𝑦−2𝐵𝐵_𝑥−2𝐵_𝑥𝑦=0,6𝐴_𝑥𝐷−3𝐵_𝑦𝐷+3𝐴𝐷_𝑥+𝐵_𝑥𝑥−2𝐶_𝑥𝑦−3𝐵𝐷_𝑦+3𝐷_𝑦𝑦+2𝐶𝐶_𝑦−𝐶𝐵_𝑥=0. (1.9) As an example we revisit the well-known modified Emden equation which has eight point symmetries [9] 𝑦^′′+3𝑦𝑦^′+𝑦³=0. (1.10) This ODE satisfies (1.8) and (1.9) and is reducible to the free particle equation 𝑦^′′=0 via the map (see [9]) 1𝑥=𝑥−𝑦,1𝑦=2𝑥²−𝑥𝑦. (1.11) Therefore in the sequel we consider the free particle equation as representative of all linearizable by point transformations scalar second-order ODEs.

In the next section we give the classifying relation for the symmetries of the first integrals of the free particle equation.

2. Symmetries of the Fundamental First Integrals

We consider the free particle equation 𝑦^′′=0 (2.1) which has the maximum number of symmetries, namely, eight given by (we list them here as we use these in what follows) 𝑋₁=𝜕𝑋𝜕𝑥₂=𝜕𝑋𝜕𝑦₃𝜕=𝑥𝑋𝜕𝑥₄𝜕=𝑦𝑋𝜕𝑦₅𝜕=𝑥𝑋𝜕𝑦₆𝜕=𝑦𝑋𝜕𝑥₇=𝑥²𝜕𝜕𝜕𝑥+𝑥𝑦𝑋𝜕𝑦₈𝜕=𝑥𝑦𝜕𝑥+𝑦²𝜕.𝜕𝑦 (2.2) It is clear that the free particle equation (2.1) has two functionally independent first integrals 𝐼₁=𝑦^′,𝐼₂=𝑥𝑦^′−𝑦. (2.3a)(2.3b)The first integral (2.3a) has three symmetries [4] 𝑋₁=𝜕𝑋𝜕𝑥₂=𝜕𝑋𝜕𝑦₃𝜕=𝑥𝜕𝜕𝑥+𝑦,𝜕𝑦 (2.4) and (2.3b) also has three symmetries [4] 𝐺₁𝜕=𝑥𝐺𝜕𝑥₂𝜕=𝑥𝐺𝜕𝑦₃=𝑥²𝜕𝜕𝜕𝑥+𝑥𝑦.𝜕𝑦 (2.5) We observe that the symmetries of the first integral of (2.3a) are the same as that of (2.3b) if we multiply the symmetries of (2.3a) by 𝑥 which is the multiplier or characteristic of the free particle equation that results in the integral 𝐼₂.

Let us see what happens if we find the symmetries of the quotient of the first integrals ((2.3a) and (2.3b)), namely, 𝐼₂𝐼₁𝑦=𝑥−𝑦^′. (2.6) As shown in [4], (2.6) possesses three symmetries as well. These are 𝑌₁𝜕=𝑦,𝑌𝜕𝑥₂𝜕=𝑦,𝑌𝜕𝑦₃𝜕=𝑥𝑦𝜕𝑥+𝑦²𝜕,𝜕𝑦 (2.7) which are the same as the symmetries (2.4) if we multiply the symmetries of (2.3a) by 𝑦. However, this is not a multiplier of our equation.

It was demonstrated, in the seminal paper [4], that the Lie algebras of the symmetries of the first integrals 𝐼₁, 𝐼₂ and their quotient 𝐼₂/𝐼₁ are isomorphic. Also each triplet (2.4), (2.5), and (2.7) can be mapped into the other by a projective transformation. Furthermore, it was noted in [4] that the three triplets together generate the Lie algebra 𝑠𝑙(3,𝑅) of the free particle equation.

3. Classifying Relation for the Symmetries

We know (see [4]) the symmetries of the functionally independent first integrals 𝐼₁ and 𝐼₂ or their quotient of the free particle equations. These are what we briefly reviewed and commented on in the previous section. Now the question arises if we want to know the symmetry properties of say the product 𝐼₁𝐼₂. We then need to compute them from first principles by using the symmetry condition. Instead of doing this each time from the beginning principles, can one obtain the relationship between the symmetries and first integrals? This is what we do here. The benefit of having such a relation enables us to also classify the first integrals of the free particle equation in terms of their point symmetries.

Let then 𝐹 be an arbitrary function of 𝐼₁ and 𝐼₂, namely, 𝐹=𝐹(𝐼₁,𝐼₂). The symmetry of this general function of the first integrals is 𝑋^[1]𝐹=𝑋^[1]𝐼₁𝜕𝐹𝜕𝐼₁+𝑋^[1]𝐼₂𝜕𝐹𝜕𝐼₂=0, (3.1) where 𝑋^[1]𝐼₁=𝜉𝜕𝜕𝜕𝑥+𝜂𝜕𝑦+𝜁_𝑥𝜕𝜕𝑦^′𝑦^′=𝜁_𝑥𝑋^[1]𝐼₂=𝜉𝜕𝜕𝜕𝑥+𝜂𝜕𝑦+𝜁_𝑥𝜕𝜕𝑦^′𝑥𝑦^′−𝑦=𝜉𝑦^′+𝑥𝜁_𝑥−𝜂. (3.2) Now 𝜉, 𝜂, and 𝜁_𝑥 are 𝜉=𝑎₁+𝑥𝑎₃+𝑦𝑎₆+𝑥²𝑎₇+𝑥𝑦𝑎₈𝜂=𝑎₂+𝑦𝑎₄+𝑥𝑎₅+𝑥𝑦𝑎₇+𝑦²𝑎₈𝜁_𝑥=−𝑦^′𝑎₃+𝑦^′𝑎₄+𝑎₅−𝑦^′2𝑎₆+𝑦−𝑥𝑦^′𝑎₇+𝑦𝑦^′−𝑥𝑦^′2𝑎₈. (3.3) These are the coefficients of 𝑋^[1] which are obtained by setting 𝑋^[1]=⁸_𝑖=1𝑎_𝑖𝑋_𝑖^[1], (3.4) where 𝑋_𝑖s are the free particle symmetries as given in (2.2), and the 𝑎_𝑖s are constants. The reason for this is that the symmetries of the first integrals are always the symmetries of the free particle equation (see [6]).

After substituting the values of 𝑋^[1]𝐼₁, 𝑋^[1]𝐼₂ as in (3.2), with 𝜉, 𝜂, and 𝜁_𝑥 as in (3.3), in (3.1), we get after some calculations −𝑦^′𝑎₃+𝑦^′𝑎₄+𝑎₅−𝑦^′2𝑎₆+𝑦−𝑥𝑦^′𝑎₇+𝑦𝑦^′−𝑥𝑦^′2𝑎₈𝜕𝐹𝜕𝐼₁+𝑎₁+𝑥𝑎₃+𝑦𝑎₆+𝑥²𝑎₇+𝑥𝑦𝑎₈𝑦^′+−𝑦^′𝑎₃+𝑦^′𝑎₄+𝑎₅−𝑦^′2𝑎₆+𝑦−𝑥𝑦^′𝑎₇+𝑦𝑦^′−𝑥𝑦^′2𝑎₈𝑥−𝑎₂+𝑦𝑎₄+𝑥𝑎₅+𝑥𝑦𝑎₇+𝑦²𝑎₈𝜕𝐹𝜕𝐼₂=0. (3.5) Then by using the relations 𝐼₁=𝑦^′ and 𝐼₂=𝑥𝑦^′−𝑦 from ((2.3a) and (2.3b)), we finally arrive at the classifying relation −𝐼₁𝑎₃+𝐼₁𝑎₄+𝑎₅−𝐼²₁𝑎₆−𝐼₂𝑎₇−𝐼₁𝐼₂𝑎₈𝜕𝐹𝜕𝐼₁+𝐼₁𝑎₁−𝑎₂+𝐼₂𝑎₄−𝐼₁𝐼₂𝑎₆−𝐼²₂𝑎₈𝜕𝐹𝜕𝐼₂=0. (3.6) The relation (3.6) provides the relationship between the symmetries and first integrals of the free particle equation. We remind the reader that any symmetry of a first integral of the free particle equation is contained in the condition (3.6). We use this to classify the first integrals according to their symmetries.

4. Symmetry Structure of First Integrals

We invoke the classifying relation (3.6) to establish the number of symmetries possessed by the first integrals of the free particle equation.

There arise four cases.

Case 1 (no symmetry). If 𝐹 is any arbitrary function of 𝐼₁ and 𝐼₂, then 𝐹_𝐼1 and 𝐹_𝐼2 are not related to each other. In this case we have from (3.6) that −𝐼₁𝑎₃+𝐼₁𝑎₄+𝑎₅−𝐼²₁𝑎₆−𝐼₂𝑎₇−𝐼₁𝐼₂𝑎₈𝐼=0,₁𝑎₁−𝑎₂+𝐼₂𝑎₄−𝐼₁𝐼₂𝑎₆−𝐼²₂𝑎₈=0. (4.1) It is easy to see from (4.1) that all the as are zero. Therefore there exists no symmetry for this case.
As an illustrative example, if we take 𝐹=𝐼₁ln𝐼₂, then (3.6) yields −𝐼₁𝑎₃+𝐼₁𝑎₄+𝑎₅−𝐼²₁𝑎₆−𝐼₂𝑎₇−𝐼₁𝐼₂𝑎₈𝐼₂ln𝐼₂+𝐼₁𝑎₁−𝑎₂+𝐼₂𝑎₄−𝐼₁𝐼₂𝑎₆−𝐼²₂𝑎₈𝐼₁=0. (4.2) This straightforwardly results in all the as being zero.
The results here are quite unexpected and surprising. One will not have imagined a zero symmetry case for a first integral of the simplest equation! The consequence of this is as follows.
If we set the first integral to be a constant as in the example, we have 𝑦^′ln𝑥𝑦^′−𝑦=𝐶. (4.3) To integrate this kind of messy integral (4.3) and find the solution of the free particle equation from it is not easy. But this difficulty is avoidable. One does not usually obtain complicated first integral such as (4.3) in ones computation in the first place by using the approaches such as the direct method, Noether’s theorem, and multiplier approach.

Case 2 (one Symmetry). Firstly we notice that if 𝐹 satisfies the classifying relation (3.6), then 𝑋, which is a linear combination of the free particle generators, is a symmetry of this classifying relation. We also observe from (3.6) that if one has any of the free symmetry generators 𝑋_𝑖 as a symmetry of a first integral of the equation, then one ends up with three symmetries! That is one can have more than one symmetry.
Say if we take 𝑎₂ arbitrary, that is, 𝑋=𝜕/𝜕𝑦, then (3.6) yields (since 𝜕𝐹/𝜕𝐼₂=0 and 𝜕𝐹/𝜕𝐼₁≠0) −𝐼₁𝑎₃+𝐼₁𝑎₄+𝑎₅−𝐼²₁𝑎₆−𝐼₂𝑎₇−𝐼₁𝐼₂𝑎₈=0, (4.4) which in turn implies that 𝑎₁ is arbitrary and 𝑎₃=𝑎₄ as well. Thus we get more than one symmetry.
We in fact arrive at the three symmetries given in (2.4). The same applies for the other symmetries taken one at a time.
However, we do have several cases when exactly one symmetry occurs.
If we take 𝐹=𝐼₁𝐼₂ or any function of the product, then the relation (3.6) gives rise to exactly the symmetry 𝜕𝑋=2𝑥𝜕𝜕𝑥+𝑦.𝜕𝑦 (4.5) If we let 𝐹=exp(𝐼²₁𝐼₂), then (3.6) results in only 𝜕𝑋=3𝑥𝜕𝜕𝑥+2𝑦.𝜕𝑦 (4.6) As another simple example, if we set 𝐹=𝐼₁exp(−𝐼₂), then (3.6) implies the one symmetry 𝜕𝑋=𝑥+𝜕𝜕𝑥.𝜕𝑦 (4.7) As a matter of interest there are infinitely many one-symmetry cases.
To see this we consider the first integral 1𝐹=2𝐼²₁−𝑎𝐼₂,𝑎≠0. (4.8) The relation (3.6) then yields −𝐼₁𝑎₃+𝐼₁𝑎₄+𝑎₅−𝐼²₁𝑎₆−𝐼₂𝑎₇−𝐼₁𝐼₂𝑎₈𝐼₁+𝐼₁𝑎₁−𝑎₂+𝐼₂𝑎₄−𝐼₁𝐼₂𝑎₆−𝐼²₂𝑎₈(−𝑎)=0. (4.9) Separation with respect to powers of 𝐼₁ and 𝐼₂ gives rise to 𝑎𝑎₁=𝑎₅. Therefore we have the parameter-dependent symmetry 𝑋=𝑋₁+𝑎𝑋₅. (4.10) Yet a more complicated one is 𝑋=𝑋₁+𝑎𝑋₅+𝑎𝑋₆,𝑎≠0. (4.11) This symmetry is associated with the first integral 𝐼𝐹=²𝑎−1²𝐼²₁𝑎−𝑎 (4.12) which can be constructed just as before. Similarly, there are many possibilities for one symmetry.
Therefore the one symmetry case is not unique. Next we discuss the two symmetry case.

Case 3 (two symmetries). We have already seen from Case 2 that the translations in 𝑥 and 𝑦 symmetries further imply the uniform scaling symmetry. Thus one cannot have two symmetries of translations alone that are associated with a first integral of the free particle equation. Likewise the same applies for the translations in 𝑦 and the uniform scaling symmetries.
Further if we have the symmetries 𝜕𝑋=𝜕𝜕𝑦,𝑌=𝑥𝜕𝑦 (4.13) which forms the two-dimensional Abelian algebra, then 𝑎₂ and 𝑎₅ are arbitrary in (3.6). This directly gives 𝜕𝐹𝜕𝐼₁=𝜕𝐹𝜕𝐼₂=0, (4.14) and hence no integral. This means that one does not have these types of symmetries admitted by any first integral of the free particle equation.
The same argument applies if we consider 𝜕𝑋=𝜕𝜕𝑦,𝑌=𝑦𝜕𝑦 (4.15) which forms a two-dimensional non-Abelian algebra. Here again this two-dimensional algebra is not admitted by any integral of the equation.
So when do two symmetries occur for a first integral of the free particle equation? From the above it is clear that the simple type of symmetry combinations does not form two symmetries of an integral. Thus there have to be combinations of the symmetries. One such combination is 𝜕𝑋=𝜕𝜕𝑥−𝑥,𝜕𝜕𝑥𝑌=𝜕𝜕𝑦−𝑥.𝜕𝑦 (4.16) The Lie algebra formed by (4.16) is two dimensional with commutator [𝑋,𝑌]=−𝑌. Here the combination of symmetries means that 𝑎₃=−𝑎₁ and 𝑎₅=−𝑎₂. The use of these in the relation (3.6) forces 𝐹 to satisfy the one condition 𝜕𝐹𝜕𝐼₁+𝜕𝐹𝜕𝐼₂=0, (4.17) which gives the independent integral 𝐹=𝐼₂−𝐼₁. (4.18) Hence this 𝐹 admits two symmetries.
We now look at a case in which at least one of the symmetries has a parameter in it. This is provided by the operators 𝑋=𝑋₁+𝑎𝑋₆,𝑎≠0𝑌=𝑋₂+𝑎𝑋₄. (4.19) The symmetries (4.19) span a two-dimensional algebra with [𝑋,𝑌]=−𝑎𝑋. (4.20) Here 𝐹 is given by 𝐼𝐹=𝑎²𝐼₁−1𝐼₁. (4.21) We conclude by saying that the two symmetry case is not unique.

Case 4 (three symmetry). We now present a detailed study of possible three-dimensional algebra of symmetries admitted by first integrals of the free particle equation. Two essential deductions come out of our analysis. Firstly we show that the three-dimensional algebra admitted by a first integral is unique. Secondly we prove that three is the maximal dimension admitted by any integral.
We utilize the realizations of three-dimensional Lie algebras in the real plane given by Mahomed and Leach [10]. However, we use the notation given in Ibragimov and Mahomed [11] (see also Mahomed [2]), that is, 𝐿^𝛼_3;𝑖, where 3 refers to the dimension of the algebra, 𝑖 to the number of the algebra in some given ordering and 𝛼 is the realization as an algebra may have more than one realization. For example, 𝐿^II_3;4 denotes the second realization of the fourth Lie algebra of dimension 3.
All canonical forms of three-dimensional real Lie algebras in the plane is given in Table 1. This is taken from [2].

Table 1 

Realizations of three-dimensional algebras in the real plane.

Remark 4.1. We point out that the Lie algebras 𝐿_3;1 and 𝐿^IV_3;8 are not admitted by any scalar second-order ODE. Hence we do not consider these hereafter (see [10]).

Instead of using the realizations 𝐿^I_3;8, 𝐿^II_3;8, and 𝐿^III_3;8 given in Table 1, we use the free particle generators (see [12]) 𝑋₁=𝑝,𝑋₂1=𝑥𝑝+2𝑦𝑞,𝑋₃=𝑥²𝑋𝑝+𝑥𝑦𝑞,₁=𝑝+𝑥𝑞,𝑋₂=𝑥𝑝+2𝑦𝑞,𝑋₃=2𝑥²𝑋−𝑦𝑝+2𝑥𝑦𝑞,₁=𝑝−𝑥𝑞,𝑋₂=−𝑥𝑝+2𝑦𝑞,𝑋₃=2𝑥²+𝑦𝑝+2𝑥𝑦𝑞. (4.22)(4.23)(4.24) Therefore the realizations of three-dimensional algebras given in Table 1 by replacement of 𝐿^I_3;8, 𝐿^II_3;8, and 𝐿^III_3;8 by their free particle operators (4.22), (4.23), and (4.24) above, except 𝐿_3;1 and 𝐿^IV_3;8, are free particle symmetry generators. We utilize these in our analysis below.

As 𝐿_3;1 is not admitted by the free particle equation, we begin with 𝐿_3;2. We want this algebra to be admitted by a first integral of the free particle equation. We utilize the classifying relation (3.6). Therefore 𝑎₁, 𝑎₂, and 𝑎₅ are arbitrary which imply that 𝐹 is constant. Hence this algebra is not admitted by any first integral.

The same applies to the algebras 𝐿^I_3;3, 𝐿^II_3;3, 𝐿^I_3;4, 𝐿^II_3;4, 𝐿^II_3;5, 𝐿^I_3;6, 𝐿^II_3;6,  𝐿^I_3;7, 𝐿^II_3;7 and 𝐿^I_3;8.

We separately consider the algebra 𝐿^II_3;8. We show that this algebra is not admitted by an integral as well. For what follows we utilize the free particle representation (4.23). We find that these operators correspond to 𝑎₁=𝑎₅, 2𝑎₃=𝑎₄, and 𝑎₆=−𝑎₇. The substitution of the latter in the relation (3.6) results in the three conditions on 𝐹, namely, 𝐼₁𝜕𝐹𝜕𝐼₁+2𝐼₂𝜕𝐹𝜕𝐼₂=0,𝜕𝐹𝜕𝐼₁+𝐼₁𝜕𝐹𝜕𝐼₂𝐼=0,₂₁−𝐼₂𝜕𝐹𝜕𝐼₁+𝐼₁𝐼₂𝜕𝐹𝜕𝐼₂=0. (4.25) The first two imply that 𝐹 is constant which satisfy the third. Thus there is no algebra of this type admitted by a first integral of the free particle equation. The analysis for 𝐿^III_3;8 is similar and this algebra too is not admitted.

For 𝐿_3;9, the operators imply that 𝑎₆=−𝑎₅, 𝑎₇=𝑎₁, and 𝑎₈=𝑎₂, the insertion of which into the relation (3.6) gives a condition on 𝐹 with 𝑎₁, 𝑎₂, and 𝑎₅ arbitrary. Then the result that 𝐹 must be constant arises. Thus this algebra is not admitted as well.

In the case of 𝐿^I_3;5 we have that 𝐼₁ has this algebra. This is precisely the algebra of the symmetries given in (2.4). We mention that the symmetries given in (2.5) and (2.7) also form the algebra 𝐿^I_3;5 as a projective transformation (see [4]) maps each of the representations to the one given in (2.4).

In conclusion, we have that the only three-dimensional algebra admitted by a first integral of the free particle equation is 𝐿^I_3;5.

We can state the following theorems.

Theorem 4.2. A first integral of the free particle or any scalar linearizable, by point transformation, second-order ODE admits a three-dimensional algebra if and only if the algebra is 𝐿^I_3;5.

The proof follows from the preceding discussion. Also we note that this algebra 𝐿^I_3;5 is admitted by the integrals 𝐼₁, 𝐼₂ or 𝐼₂/𝐼₁.

Theorem 4.3. The maximum dimension of the algebra admitted by any first integral of the free particle or any scalar linearizable, by point transformation, second-order ODE is three, and the algebra is 𝐿^I_3;5.

Proof. A first integral of the free particle or scalar second-order ODE, which is linearizable by point transformation, cannot admit a maximal algebra of dimension more than three since the functionally independent integrals or their quotient has the unique three-dimensional algebra which corresponds to 𝐿^I_3;5. The other integrals possess lower-dimensional algebras.

5. Concluding Remarks

In this work we have provided the algebraic structure of first integrals of the free particle or any scalar linearizable, via point transformation, ODE. Firstly, we derived the relationship between the symmetries and the first integrals of the free particle equation. By analyzing this classifying relation (3.6), we were able to establish the number of symmetries possessed by any first integral of the free particle equation. We obtained the important result that the symmetries admitted by a first integral can be 0, 1, 2, or 3. It was observed that the zero symmetry case was rather surprising or unexpected as one does not have a route to the integration of the equation due to the lack of any symmetry and this too for the simplest equation. The one- and two-symmetry cases were not unique; there were many first integrals with differing one- and two-symmetry structures. These were carefully discussed. Finally, we studied completely the situation when a first integral has three symmetries. We used the classification of realizations in the plane adapted as free particle symmetries. We showed that the only three-dimensional algebra admitted by a first integral of the free particle equation is 𝐿^I_3;5 which is admitted by the functionally independent integrals 𝐼₁ and 𝐼₂ as well as their quotient 𝐼₂/𝐼₁. Although this triplet of symmetries was discovered before in the seminal work of Leach and Mahomed [4], these authors did not prove that it was unique nor the maximum algebra. However, they did emphasize the important result that the algebras of the triplets of symmetries were isomorphic and that the three triplets of symmetries generate the 𝑠𝑙(3,𝑅) symmetry of the equation. We showed that the maximum algebra is indeed the three-dimensional algebra 𝐿^I_3;5 by completely analyzing all representations of the three-dimensional algebras.

Acknowledgments

K. S. Mohamed thanks the University of the Witwatersrand and the NRF of South Africa for financial support. E. Momoniat is grateful to the NRF for a research grant.