1. Introduction
An invertible point transformation that maps every element in a family ℱ of differential equations of a specified form into the same family is commonly referred to as an equivalence transformation of the equation [1–3]. Elements of the family ℱ are generally labeled by a set of arbitrary functions, and the set of all equivalence transformations forms, in general, an infinite dimensional Lie group called the equivalence group of ℱ. One type of equivalence transformations usually considered [1, 4, 5] is that in which the arbitrary functions are also transformed. More specifically, if we denote by A=(A1,…,Am) the arbitrary functions specifying the family element in ℱ, then for given independent variables x=(x1,…,xp) and dependent variable y, this type of equivalence transformations takes the form(1.1a)x=φ(z,w,B),(1.1b)y=ψ(z,w,B),(1.1c)A=ζ(z,w,B),where z=(z1,…,zp) is the new set of independent variables, w=w(z) is the new dependent variable, and B=(B1,…,Bm) represents the new set of arbitrary functions. The original arbitrary functions Ai may be functions of x, y, and the derivatives of y up to a certain order, although quite often they arise naturally as functions of x alone, and for the equivalence transformations of the type (1.1a), (1.1b), and (1.1c), the corresponding equivalence group that we denote by GS is simply the symmetry pseudogroup of the equation, in which the arbitrary functions are also considered as additional dependent variables.

The other type of equivalence transformations commonly considered [2, 6–8] involves only the ordinary variables of the equation, that is, the independent and the dependent variables, and thus with the notation already introduced, it consists of point transformations of the form(1.2a)x=φ(z,w),(1.2b)y=ψ(z,w).If we let G denote the resulting equivalence group, then it follows from a result of Lie [9] that G induces another group of transformations Gc acting on the arbitrary functions of the equation. The invariants of the group Gc are what are referred to as the invariants of the family ℱ of differential equations, and they play a crucial role in the classification and integrability of differential equations [1, 6, 10–14].

In the recent scientific literature, there has been a great deal of interest for finding infinitesimal methods for the determination of invariant functions of differential equations [2, 7, 15–17]. Some of these methods consist in finding the infinitesimal (generic) generator X of GS, and then using it in one way or another [7, 18] to obtain the infinitesimal generator X0 of Gc, which gives the determining equations for the invariant functions. Most of these methods remain computationally demanding and in some cases quite inefficient, perhaps just because the connection between the three groups G, Gc, and GS does not seem to have been fully investigated.

We therefore present in this paper a comparison of the groups G and GS and show in particular that G can be identified with a subgroup of GS, and we exhibit a case where the two groups are isomorphic. We also show that the generator X of GS admits a simple linear decomposition of the form X=X1+X2, where X1 is an operator uniquely associated with G, and we also give a very simple and systematic method for extracting X1 from X. This decomposition also turns out to be intimately associated with the Lie algebraic structure of the equation, as we show that X1 and X2 each generate a Lie algebra, the two of which are closely related to the components of the Levi decomposition of the Lie algebra of GS.

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We will call type I the equivalence transformations of the form (1.2a) and (1.2b) and type II those of the form (1.1a), (1.1b), and (1.1c), whose equivalence groups we have denoted by G and GS, respectively. When the coordinates system in which a vector field is expressed is clearly understood, it will be represented only by its components, so that a vector field
(2.1)ω=ξ∂x+η∂y+ϕ∂A
will be represented simply by ω={ξ,η,ϕ}. On the other hand, for a vector a=(a1,…,an) representing a subset of coordinates, the notation f∂a will mean
(2.2)f∂a=f1∂a1+⋯+fn∂an,f=(f1,…,fn).
Hence, with the notation introduced in the previous section, we may represent the generator X of GS as
(2.3)X={ξ,η,ϕ}≡ξ∂x+η∂y+ϕ∂A.
Let V={ξ,η} be the projection of this generator into the (x,y)-space, and let V0={ξ0,η 0} be the infinitesimal generator of G. Elements of ℱ may be thought of as differential equations of the form
(2.4)Δ(x,y(n);A1,…,Am)=0,
where y(n) denotes y and all its derivatives up to the order n. We have the following result.

Theorem 2.1.
(a) The group G can be identified with a subgroup of GS.

(b) The component functions ξ0 and η0 are particular values of the functions ξ and η, respectively.

Proof.
Suppose that the action of Gc induced by that of G on the arbitrary functions of the equation is given by the transformations
(2.5)Ai=γi(z,w,B1,…,Bm), i=1,…,m.
Then, since (1.2a) and (1.2b) leave the equation invariant except for the arbitrary functions, by also viewing the functions Ai as dependent variables, (1.2a) and (1.2b) together with (2.5) constitute a symmetry transformation of the equation. This is more easily seen if we consider the inverse transformations of (1.2a) and (1.2b) which may be put in the form(2.6a)z=φ^(x,y),(2.6b)w=ψ^(x,y).

If we now denote by
(2.7)Bi=γ^(x,y,A1,…,Am), i=1,…,m
the resulting arbitrary functions in the transformed equation, it follows that in terms of the new set of variables z, w, and Bi, any element of ℱ is locally invariant under (2.6a), (2.6b), and (2.7), and this proves the first part of the theorem. The second part of the theorem is an immediate consequence of the first part, for we can associate with any element (φ,ψ) of G a triplet (φ,ψ,γ) in GS, where γ is the action in (2.5) induced by (φ,ψ) on the arbitrary functions of the equation. The result thus follows by first recalling that GS has generic generator X={ξ,η,ϕ} and by considering the infinitesimal counterpart of the finite transformations (φ,ψ,γ), which must be of the form {ξ0,η0,ζ0} for a certain function ζ0.

On the basis of Theorem 2.1, it is clear that one can obtain the generator V0={ξ0,η0} of G by imposing on the projection V={ξ,η} of X the set of minimum conditions Ω that reduces it to the infinitesimal generator of the equivalence group G of ℱ, so that V0=V|Ω. It was also observed (see [7]) that in case A is the function of x alone, if we let ϕ0 denote the resulting value of ϕ when these minimum conditions are also imposed on X={ξ,η,ϕ}, then the generator X0 of Gc can be obtained by setting X0={ξ0,ϕ0}. However, the problem that arises is that of finding the simplest and most systematic way of extracting X1=X|Ω={ξ0,η0,ϕ0} from X={ξ,η,ϕ}.

To begin with, we note that the coefficient ϕ0 is an m-component vector that depends in general on (p+1)+m variables, and finding its corresponding finite transformations by integrating the vector field {ξ0,η0,ϕ0} can be a very complicated task. Fortunately, once the finite transformations of the generator V0 of G which are easier to find are known, we can easily obtain those associated with ϕ0 using the following result.

Lemma 2.2.
The finite transformations associated with the component ϕ0 of X1={ξ0,η0,ϕ0} are precisely given by the action (2.7) of Gc induced by that of (2.6a) and (2.6b).

Proof.
Since X1=X|Ω, where Ω is the set of minimum conditions to be imposed on V={ξ,η} to reduce it into an infinitesimal generator V0={ξ0,η0} of G, it first follows that once the finite transformations (2.6a) and (2.6b) corresponding to V0 are applied to the equation, the resulting equation is invariant, except for the expressions of the arbitrary functions which are now given by (2.7). Thus if (z,w,b) are the new variables generated by the symmetry operator X1, where b=(b1,…,bm), then the only way to have an invariant equation is to set
(2.8)bi=γ^(x,y,A1,…,Am), i=1,…,m,
where γ^ is the same function appearing in (2.7), and this readily proves the lemma.

3. Case of the General Third Order Linear ODE
We will look more closely at the connection between the two operators X and X1 by considering the case of the family of third-order linear ordinary differential equations (ODEs) of the form
(3.1)y(3)+a1(x)y'+a0(x)y=0,
which is said to be in its normal reduced form. Here, the arbitrary functions Ai of the previous section are simply the coefficients aj of the equation. This form of the equation is in no way restricted, for any general linear third order ODE can be transformed into (3.1) by a simple change of the dependent variable [8, 16]. If we consider the arbitrary functions aj as additional dependent variables, then by applying known procedures for finding Lie point symmetries [19–21], the infinitesimal generator X of the symmetry group GS in the coordinates system (x,y,a1,a0) is found to be of the form(3.2a)X={f,(k1+f′)y+g,-2(a1f′+f(3)),C4},
where
(3.2b)C4=-1y(a0g+a1g′+g(3))-(3a0f′+a1f′′+f(4)),and where f and g are arbitrary functions of x. The projection of X in the (x,y)-space is therefore
(3.3)V={f,(k1+f′)y+g},
and a simple observation of this expression shows that due to the homogeneity of (3.1), (3.3) may represent an infinitesimal generator of the equivalence group G only if g=0. A search for the one-parameter subgroup exp(tW), satisfying exp(tW)(x,y)=(z,w) and generated by the resulting reduced vector field W={f,(k1+f')y}, readily gives
(3.4)z˙=f(z),w˙=(k1+f′(z))w,
where
(3.5)z˙=dzdt, w˙=dwdt.
Integrating these last two equations while taking into account the initial conditions gives(3.6a)J(z)=t+J(x),(3.6b)w=ek1tf(z)f(x)y,
where
(3.6c)J(z)=∫dzf(z).Differentiating both sides of (3.6a) with respect to x shows that dz/dx=f(z)/f(x). Thus, if we assume that z is explicitly given by
(3.7)z=Ft(x)≡F(x),
for some function F, then this leads to(3.8a)z=F(x),(3.8b)w=ek1tF′(x)y,and we thus recover the well-known equivalence transformation [6, 8, 14] of (3.1). Therefore, the condition g=0 is the necessary and sufficient condition for the vector V in (3.3) to represent the infinitesimal generator of G. In other words, the set Ω of necessary and sufficient conditions to be imposed on X to obtain X1 is reduced in this case to setting g=0. More explicitly, we have(3.9a)V0={f,(k1+f′)y}≡{ξ0,η0},(3.9b)ϕ0={-2(a1f′+f′′′),-(3a0f'+a1f′′+f(4))},(3.9c)X1={f,(k1+f′)y,-2(a1f′+f′′′),-(3a0f′+a1f′′+f(4))},where X1=X|g=0.

We would now like to derive some results on the algebraic structure of LS, the Lie algebra of the group GS related to (3.1), and its connection with that for the corresponding group G. Thus, for any generator X of GS, set X2=X-X1, where X1=X|g=0 is given by (3.9c), while X2 takes the form
(3.10)X2={0,g,0,-1y(a0g+a1g'+g(3))}.

Since X1 depends on f and k1 while X2 depends on g, we set (3.11a)X1(f,k1)={f,(k1+f′)y,-2(a1f′+f′′′),-(3a0f′+a1f′′+f(4))},(3.11b)X2(g)={0,g,0,-1y(a0g+a1g'+g(3))},for any arbitrary functions f and g and arbitrary constant k1. Let L0, L1, and L2 be the vector spaces generated by X1(f,0), X1(f,k1), and X2(g), respectively, where f, g, and k1 are viewed as parameters. Let
(3.12)LS,0=L0∔L2
be the subspace of the Lie algebra LS=L1∔L2 of GS. We note that LS,0 is obtained from LS simply by setting k1=0 in the generator X1(f,k1) of GS, which according to (3.8b) amounts to ignoring the constant factor λ=ek1t in the transformation of the dependent variable under G. Moreover, we have dim LS,0=dim LS-1, while LS itself is infinite dimensional, in general.

Theorem 3.1.
(a) The vector spaces L0, L1, and L2 are all Lie subalgebras of LS.

(b) L0 and L2 are the components of the Levi decomposition of the Lie algebra LS,0, that is,
(3.13) [L0,L2]⊂L2,
and L2 is a solvable ideal while L0 is semisimple.

Proof.
A computation of the commutation relations of the vector fields shows that(3.14a)[X1(f1,k1),X1(f2,k2)]=X1(-f2f1′+f1f2′,0),(3.14b)[X2(g1),X2(g2)]=0,(3.14c)[X1(f1,k1),X2(g1)]=X2(f1g1′-g1(k1+f1′)),where the fj and gj are arbitrary functions, while the kj are arbitrary constants. Consequently, it readily follows from (3.14a) and (3.14b) that L1 and L2 are Lie subalgebras of LS, while setting k1=k2=0 in (3.14a) shows that L0 is also a Lie subalgebra, and this proves the first part of the theorem. Moreover, it follows from the commutation relations (3.14a), (3.14b), and (3.14c) that [LS,LS]⊂LS,0, and hence that LS,0 is an ideal of LS, while (3.14b) and (3.14c) show that L2 is an abelian ideal in LS, and in particular in LS,0. Thus, we are only left with showing that L0 is a semisimple subalgebra of LS,0. Clearly, [L0,L0]≠0, and if L0 had a proper ideal A, then for a given nonzero operator X1(H,0) in A, all operators X1(-fH′+f′H,0) would be in A for all possible functions f. However, since for every function h of x the equation
(3.15)-fH'+f'H=h
admits a solution in f, it follows that A would be equal to L0. This contradiction shows that L0 has no proper ideal and is therefore a simple subalgebra of LS,0.

Note that part (b) of Theorem 3.1 can also be interpreted as stating that up to a constant factor, X1 and X2 generate the components of the Levi decomposition of LS. The theorem therefore shows that the decomposition X=X1+X2 is not fortuitous, as it is intimately associated with the the Levi decomposition of LS, and this decomposition is unique up to isomorphism for any given Lie algebra.

Although we have stated the results of this theorem only for the general linear third order equation (3.1) in its normal reduced form, these results can certainly be extended to the general linear ODE
(3.16)y(n)+an-1y(n-1)+an-2y(n-2)+⋯+a0y=0
of an arbitrary order n≥3. We first note that if we write the infinitesimal generator X of the symmetry group GS of this equation in the form
(3.17)X={ξ,η,ϕ}≡ξ∂x+η∂y+ϕ∂A,
where A={an-1,an-2,…,a0} is the set of all arbitrary functions, then on account of the linearity of the equation, we must have
(3.18)η=hy+g
for some arbitrary functions h and g. Now, let again X1={ξ0,η0,ϕ0} and X2 be given by
(3.19)X1=X|g=0, X2=X-X1,
and set X0={ξ0,ϕ0}. We have shown in another recent paper [16] that X0 thus obtained using g=0 as the minimum set of conditions is the infinitesimal generator of the group Gc for n=3,4,5. This should certainly also hold for the linear equation (3.16) of a general order, and we thus propose the following.

Conjecture 3.2.
For the general linear ODE (3.16), X0={ξ,ϕ}|g=0 is the infinitesimal generator of Gc, where X={ξ,η,ϕ} is the generator of GS.

As already noted, it has been proved [7] that for any family ℱ of (linear or nonlinear) differential equations of any order in which the arbitrary functions depend on the independent variables alone, if X1={ξ0,η0,ϕ0} is obtained by setting X1=X|Ω for some set Ω of minimum conditions that reduce V={ξ,η} into a generator of G, then X0={ξ0,ϕ0} is the generator of Gc. However, the difficulty lies in finding the set Ω of minimum conditions, and we have proved that for (3.1), Ω is given by {g=0} and extended this as a conjecture for a general linear homogeneous ODE.

Moreover, calculations done for equations of low order up to five suggest that all subalgebras appearing in Theorem 3.1 can also be defined in a similar way for the general linear equation (3.16) and that all the results of the theorem also hold for this general equation.

We now wish to pay some attention to the converse of part (a) of Theorem 2.1 which states that for any given family ℱ of differential equations, the group G can be identified with a subgroup of GS. From the proof of that theorem, it appears that the symmetry group GS is much larger in general, because there are symmetry transformations that do not arise from type I equivalence transformations. A simple example of such a symmetry is given by the term X2 appearing in (3.10) of the symmetry generator of (3.1). Indeed, by construction, its projection X2,0={0,g} in the (x,y)-space does not match any particular form of the generic infinitesimal generator V0={f,(k1+f')y} of G, where f is an arbitrary function and k1 an arbitrary constant.

Nevertheless, although (3.1) gives an example in which the inclusion G⊂GS is strict, there are equations for which the two groups are isomorphic. Such an equation is given by the nonhomogeneous version of (3.1) which may be put in the form
(3.20)y(3)+a1(x)y'+a0(x)y+r(x)=0,
where r is also an arbitrary function, in addition to a1 and a0. The linearity of this equation forces its equivalence transformations to be of the form
(3.21)x=f(z), y=h(z)w+g(z),
and the latter change of variables transforms (3.20) into an equation of the form
(3.22)w′′′+B2w′′+B1w'+B0w+B-1=0,
where the Bj, for j=-1,…,2 are functions of z and
(3.23)B2=3(h′h-f′′f′).
The required vanishing of B2 shows that the necessary and sufficient condition for (3.21) to represent an equivalence transformation of (3.20) is to have h=λf' for some arbitrary constant λ. The equivalence transformations of (3.20) are therefore given by
(3.24)x=f(z), y=λf′(z)w+g(z).
On the other hand, the generator X of the symmetry group GS of the nonhomogeneous equation (3.20) in the coordinates system (x,y,a1,a0,r) is found to be of the form(3.25a)X={J,(k1+J′)y+P,-2(a1J′+J′′′),C3,ϕ4},
where
(3.25b)C3=-1y(a0P+ϕ4+2rJ'-rk1+a1P'+P(3))-(3a0J′+a1J′′+J(4)),and where J and P are arbitrary functions of x and k1 is an arbitrary constant, while ϕ4 is an arbitrary function of x,y,a1,a0, and r. Thus, X has projection V={J,(k1+J')y+P} on (x,y)-space and this is exactly the infinitesimal transformation of (3.24). Consequently, the minimum set Ω of conditions to be imposed on V to reduce it into the infinitesimal generator V0={ξ0,η0} of G is void in this case, and hence
(3.26)X=X1={ξ0,η0,ϕ0}.
It thus follows from Lemma 2.2 that the finite transformations associated with X are given precisely by (3.24), together with the corresponding induced transformations of the arbitrary functions a1,a0, and r. Consequently, to each symmetry transformation X in GS, there corresponds a unique equivalence transformation in G and vice versa. We have thus proved the following results.

Proposition 3.3.
For the nonhomogeneous equation (3.20), the groups G and GS are isomorphic.

This proposition should certainly also hold for the nonhomogeneous version of the general linear equation (3.16) of an arbitrary order n. In such cases, invariants of the differential equation are determined simply by searching the symmetry generator X of GS, which must satisfy (3.26) and then solving the resulting system of linear first-order partial differential equations (PDEs) resulting from the determining equation of the form
(3.27)X0,m·F=0,
where X0,m is the generator X0={ξ0,ϕ0} of Gc prolonged to the desired order m of the unknown invariants F.