Algebraic Properties of First Integrals for Scalar Linear Third-Order ODEs of Maximal Symmetry

and Applied Analysis 3 which as is well known has the seven symmetries (Lie [1] and e.g., [4])


Introduction
Ordinary differential equations (ODEs) are a fertile area of study, especially the Lie algebraic properties of such equations and their first integrals have great importance.Initial investigations since the works of Lie [1] were motivated by physical problems, such as the free particle, one-dimensional harmonic oscillator, and Emden-Fowler equations and classification.Indeed, most of the earlier works dealt with secondorder linear equations (see, e.g., [2][3][4][5][6][7][8]).In the investigation by Mahomed and Leach [9], they found that the Lie point symmetries of the maximal cases of scalar linear th-order ODEs,  ≥ 3, are  + 1,  + 2, and  + 4. Thus for scalar linear third-order equations these correspond to 4, 5, and 7 symmetries.Moreover for scalar third-order linear ODEs, Govinder and Leach [10] provided the algebraic structure of the basic first integrals for such equations.They showed that the three equivalence classes each has certain first integrals with a specific number of point symmetries.They followed on the initial investigation of Leach and Mahomed [11] who considered the point symmetries of the basic first integrals of linear second-order ODEs.Then in the work [12] Flessas et al. attempted the symmetry structure for the first integrals of higher-order equations of maximal symmetry.
The subject of the present paper is the investigation of the Lie algebraic properties of first integrals of scalar linear third-order ODEs of the maximal class which is represented by   = 0. We remind the reader that for the simplest class there has been some analysis made in Flessas et al. [12].This is in regards to the maximal algebra possessed by an integral of   = 0 which is listed in Table II of [12].However, this is incomplete.We extend this study and provide a complete analysis on the Lie point symmetries and first integrals for the simplest third-order ODE including the maximal algebra case.We firstly deduce the classifying relation between the point symmetries and first integrals for this simple class.Then we use this to study the point symmetry properties of the first integrals of   = 0 which also represents all linearizable by point transformations third-order ODEs that reduce to this class.
We begin by noting the condition for symmetries of the first integrals of scalar linear ODEs of order one.Then for completeness we review briefly the results of the paper by Mahomed and Momoniat [13] which discusses the relationship between the point symmetries of the first integrals of scalar linear second-order ODEs.These two cases are shown to be distinct in terms of their algebraic properties of their integrals when compared to higher-order ODEs of maximal symmetry.
1.1.Linear First-Order Equations.Consider the simplest firstorder ODE It is easy to see that is a point symmetry generator of (1) if where with in which   is the total differentiation operator and  [1] is the first prolongation of the generator .We quickly see that where  is an arbitrary function of .Therefore, Thus there is an infinite number of point symmetries.We now show that only  = (, )/ are symmetries of the first integral.This forms an infinite-dimensional subalgebra of the Lie algebra of (1).Now  =  is a first integral of (1).It has point symmetry  as in (2) if This implies that  = 0 which immediately results in There is an infinite number of symmetries of the first integral  =  of (1).
Let  be an arbitrary function of , namely,  = ().The symmetry of this general function of the first integral is Therefore  as in ( 9) is a symmetry of  =  and also any function of ().Since any scalar first-order ODE is equivalent to the simplest one (1), this means that a first integral of a nonlinear first-order ODE has infinitely many symmetries too.
As an example, we consider the nonlinear first-order Riccati equation the first integral of which is A symmetry of the first integral (12) is In fact, we have an infinite number of symmetries given by where  is an arbitrary function in its arguments.Therefore we note here that the symmetries of the first integrals of a first-order equation form a proper subalgebra of the Lie algebra of the equation itself.We cannot generate the full algebra as is the case for linear scalar second-order ODEs [11] by use of the algebra of the basic integral alone.
The symmetries of the first integrals of scalar linear second-order ODEs have interesting properties [11,13].The first integrals of such linear equations can have 0, 1, 2, and the maximum 3 symmetries [13].The Lie algebra of the maximum case is unique.Peculiar to such equations is the other remarkable property that their Lie algebra is generated by the symmetry properties of the basic integrals and their quotient [11].
Below we study the symmetry properties of first integral for the simplest scalar linear third-order ODEs of maximal point symmetry.In the case of the basic first integrals, the algebraic properties are known from the work [10].Here we pursue the relationship between symmetries and first integrals of scalar linear third-order ODEs for the simplest and maximal class.We obtain the classifying relation for this class and invoke this to arrive at counting theorems and the result on the maximal case of symmetries of the first integrals.
In the following we look at algebraic properties of first integrals for the seven point symmetry case by deriving the classifying relation between the symmetries and their first integrals.We use this relation to arrive at interesting properties which appear for the first time in the literature.

Algebraic Properties of the First Integrals of 𝑦 󸀠󸀠󸀠 = 0
We consider the simplest third-order ODE which as is well known has the seven symmetries (Lie [1] and e.g., [4]) We have listed the symmetries in the order of the solution symmetries being first, then the homogeneity symmetry, and the remaining three which form the algebra (2, ).This ODE ( 15) also represents all linearizable third-order ODEs reducible to it via point transformation.The order in which the symmetries appear in ( 16) is used in what follows.It is obvious that ( 15) has three functionally independent first integrals We use the ordering of the integrals as given in [10].The first integral (17a) has four symmetries [10] from which we observe that there are two solution symmetries, one translation in  symmetry and a scaling symmetry.The translation in  symmetry is a subset of the (2, ) symmetries with  4 being a combination of the uniform scaling symmetry in both variables contained in the (2, ) symmetries together with the homogeneity symmetry.Part of this fact was also noted in [12].The second first integral (17b) has three symmetries [10] with two solution symmetries and  3 being part of the (2, ) symmetries.The third first integral (17c) also has four symmetries [10] (20) Again one can see the solution symmetries, scaling, and the symmetry  4 which is contained in the (2, ) symmetries.Note further that the symmetries in (20) are found by multiplying those of (18) by the factor .In fact these two sets are equivalent via a point transformation [10].The other important properties are discussed in the next section.
Below we obtain the classifying relation.

2.1.
Classifying Relation for the Symmetries and Integrals.Now let  be an arbitrary function of the integrals (17a), (17b), and (17c),  1 ,  2 , and  3 , namely, The symmetry of this general function of the first integrals is  [2]  =  [2]  1   1 +  [2]  2   2 +  [2] where The coefficient functions , ,   , and   are Abstract and Applied Analysis These are obtained by setting where the   are the symmetry generators as given in ( 16) and the   are constants.The reason being that the symmetries of the first integrals are always the symmetries of the equation (see [14]).
After substitution of the values of  [2]  1 ,  [2]  2 , and  [2]  3 as in (23), with , ,   , and   given in (24), as well as by use of the first integrals  1 =   ,  2 =   −   , and 22), we arrive at the classifying relation The relation ( 26) explicitly provides the relationship between the symmetries and the first integrals of the simple thirdorder equation (15).We invoke this to classify the first integrals according to their symmetries in what follows.
We use the classifying relation (26) to establish the number and property of symmetries possessed by the first integrals of the simplest third-order equation (15).
There arise five cases.We deal with each below.
Case 1 (no symmetry).If  is any arbitrary function of  1 ,  2 , and  3 , then   1 ,   2 , and   3 are not related to each other.In this case we have from (26) that It is easy to see from ( 27) that all the 's are zero.Therefore there exists no symmetry for this case.
As an illustrative example, if we take  =  1  2 ln  3 , then (26) straightforwardly yields This easily results in all the 's being zero.
Case 2 (one symmetry).If  satisfies the relation (26), then there exists one symmetry.For the simple symmetries of (15) one obtains further symmetries except for  6 which we consider below.
In fact similar to the free particle equation [13], there are many one symmetry cases.
Case 3 (two symmetries).Here there are many cases as well.We begin by utilizing the Lie table for the classification of the two-dimensional algebras in the plane which are given, for example, in [6].They are These form subalgebras of the Lie algebra of symmetries of (15) as can clearly be observed.We take the first realization listed above.If  1 is arbitrary in (26), then  1 = / implies that  is independent of  3 .Further  5 = / yields that  does not depend on  2 as well.Since we require that / 1 ̸ = 0, we have from which it follows that  3 =  7 = 0 and  4 =  6 .Thus we end up with two more symmetries  2 and  4 +  6 .These turn out to be the four symmetries of the integral  1 given in (18).Likewise for the second realization we obtain (18) again.For the third realization listed above we find the symmetries of  2 as in (19).
Hence the first three realizations listed do not provide maximal two symmetries for the first integrals of (15).
In fact the fourth realization results in a two symmetry case as  1 and  4 are arbitrary and give rise to which has the solution The further substitution of this form into the relation ( 26) constrains all the 's to be zero except for  1 and  4 .This result prompts the following simple products and quotients that do give two symmetries.
If  =  1  2 , then relation ( 26) yields Here we observe that  2 ,  3 ,  5 , and  7 are zeros whereas  1 is arbitrary and  6 = 2 4 , and therefore we obtain the two symmetries which form a two-dimensional algebra with If we set  =  1  3 , then we end up getting  1 ,  3 ,  4 ,  5 , and  7 equal to zero.Since  2 and  6 are arbitrary so they result in two symmetries with Lie bracket If we take  =  2  3 , then we see that  3 is arbitrary and  6 = −2 4 which then give rise to the two symmetries with Consider now  =  3 / 1 .This shows that  2 and  4 are arbitrary, and the resulting two symmetries are with commutation relation If we let  =  3 / 2 , then here  3 and  4 are arbitrary, and therefore the two symmetries are with ,  1 =   , and  = / 1 (cf.[12]), then we have from the relation ( 26) This results in  1 ,  2 ,  3 , and  7 being zero whereas  5 is arbitrary, and  4 = − 6 , which give rise to the two symmetries with and  = / 3 (cf.[12]), then relation (26) yields The above relation shows that  1 ,  2 ,  3 , and  5 are zeros,  7 is arbitrary, and  4 =  6 .This imply two symmetries together with There are thus many two symmetry cases.One could obtain more.Also they could arise as different combinations of the seven symmetries (16).So one concludes that the twodimensional algebra cases are not unique.We have seen the occurrence of both Abelian and non-Abelian Lie algebras.
Case 4 (three symmetries).Here we use the three-dimensional real realizations of Mahomed and Leach [7].The notation used is that given in [4].Since we adapt these realizations as symmetries of third-order equations, the entries for the first entry  3;1 and the nonsolvable algebra  I 3;8 are those which are symmetries of such equations.
Note that for  II 3;8 ,  III 3;8 , and  IV 3;8 one can use the realizations as given in Table 1 or the ones obtained by interchanging  and  in the realizations given.The reason for this is that one still obtains third-order representative equations for the latter realizations.
As  3;1 is the first three-dimensional algebra in Table 1, we start with that.We want this Abelian algebra to be admitted by a first integral of (15).We utilize the classifying relation (26).Therefore  1 ,  2 , and  3 are arbitrary which imply that  is constant.Hence this algebra is not admitted by any first integral of ( 15) although, if it is admitted by a nonlinear thirdorder ODE, it implies linearization (see Mahomed and Leach [6]).The same applies to  II 3;6 ,  = 1/2.For both these cases, the algebras  I 3;6 ,  = 2 and  II 3;6 ,  = 1/2 are admitted by a first integral  = ( 1 ) but these are not maximal and contained in a four-dimensional algebra, spanned by operators (18).Now we focus on the three symmetry case which is admitted by  = ( 2 ).These three symmetries are given in (19).The Lie algebra of the generators (19) has nonzero commutators with the elements thus forming the Lie algebra  II 3;6 ,  = −1.In fact the transformation maps it to the canonical form of Table 1, namely, Therefore the symmetries of  = ( 2 ) has the Lie algebra  II 3;6 ,  = −1.
There is yet another three-dimensional algebra which is admitted by a first integral of (15).This occurs for  I 3;8 .We use    (15) as these are subalgebras of the maximal six-dimensional algebras (see Ibragimov and Mahomed [3]) which are admitted by nonlinear third-order ODEs not reducible to the simplest equation (15).
In the case of the algebra  3;9 one has the situation that this algebra is not a subalgebra of the seven-dimensional algebra of (15) (Wafo Soh and Mahomed [15]).
In conclusion of this discussion, we have two threedimensional algebras admitted by a first integral of (15) which are  I 3;8 and  II 3;6 ,  = −1.
Hence the picture is quite distinct for the manner in which the full Lie algebra is generated for the ODEs   = 0,   = 0 and,  () = 0,  ≥ 3.This is also consistent with the properties of their symmetry algebra which are different (see, e.g., [4]).