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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 482963, 32 pages
Research Article

Automorphisms of Ordinary Differential Equations

Department of Mathematics, Faculty of Civil Engineering, Brno University of Technology, Veveří 331/95, 602 00 Brno, Czech Republic

Received 26 July 2013; Accepted 4 October 2013; Published 28 January 2014

Academic Editor: Josef Diblik

Copyright © 2014 Václav Tryhuk and Veronika Chrastinová. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The paper deals with the local theory of internal symmetries of underdetermined systems of ordinary differential equations in full generality. The symmetries need not preserve the choice of the independent variable, the hierarchy of dependent variables, and the order of derivatives. Internal approach to the symmetries of one-dimensional constrained variational integrals is moreover proposed without the use of multipliers.

1. Preface

The theory of symmetries of determined systems (the solution depends on constants) of ordinary differential equations was ultimately established in Lie’s and Cartan’s era in the most possible generality and the technical tools (infinitesimal transformations and moving frames) are well known. Recall that the calculations are performed in finite-dimensional spaces given in advance and the results are expressed in terms of Lie groups or Lie-Cartan pseudogroups.

We deal with underdetermined systems (more unknown functions than the number of equations) of ordinary differential equations here. Then the symmetry problem is rather involved. Even the system of three first-order quasilinear equations with four unknown functions (equivalently, three Pfaffian equations with five variables) treated in the famous Cartan’s article [1] and repeatedly referred to in actual literature was not yet clearly explained in detail. Paradoxically, the common tools (the calculations in given finite-order jet space) are quite sufficient for this particular example. We will later see that they are insufficient to analyze the seemingly easier symmetry problem of one first-order equation with three unknown functions (alternatively, two Pfaffian equations with five variables) in full generality since the order of derivatives need not be preserved in this case and the finite-order jet spaces may be destroyed. Recall that even the higher-order symmetries (automorphisms) of empty systems of differential equations (i.e., of the infinite order jet spaces without any additional differential constraints) are nontrivial [24] and cannot be included into the classical Lie-Cartan theory of transformation groups. Such symmetries need not preserve any finite-dimensional space and therefore the invariant differential forms (the Maurer-Cartan forms, the moving coframes) need not exist.

Let us outline the very core of the subject for better clarity by using the common jet terminology. We start with the higher-order transformations of curves lying in the space with coordinates . The transformations are defined by certain formulae where the -smooth real-valued functions depend on a finite number of the familiar jet variables The resulting curve again lying in appears as follows. We put and assuming there exists the inverse function which provides the desired result One can also easily obtain the well-known prolongation formula for the derivatives by using the Pfaffian equations Functions satisfying (4) and may be arbitrary here.

At this place, in order to obtain coherent theory, introduction of the familiar infinite-order jet space of -parametrized curves briefly designated as with coordinates is necessary. Then formulae ((1), (6)) determine a mapping , a morphism of the jet space . If the inverse given by certain formulae exists, we speak of an automorphism (in alternative common terms, symmetry) of the jet space . It should be noted that we tacitly deal with the local theory in the sense that all formulae and identities, all mappings, and transformation groups to follow are in fact considered only on certain open subsets of the relevant underlying spaces which is not formally declared by the notation. Expressively saying, in order to avoid the clumsy purism, we follow the reasonable 19th century practice and do not rigorously indicate the true definition domains.

After this preparation, a system of differential equations is traditionally identified with the subspace given by certain equations (We tacitly suppose that is a “reasonable subspace” and omit the technical details.) This is the infinitely prolonged system. The total derivative vector field defined on is tangent to the subspace and may be regarded as a vector field on , as well. The morphism transforms into the subspace given by the equations This is again a system of differential equations. In our paper, we are interested only in the particular case when . Then, if the inverse locally exists on a neighbourhood of the subspace in the total jet space, we speak of the external symmetry   of the system of differential equations (9). Let us, however, deal with the natural restriction of the mapping to the subspace . If there exists the inverse of the restriction, we speak of the internal symmetry. Internal symmetries do not depend on the localizations of in . More precisely, differential equations can be introduced without any reference to jet spaces and the internal symmetries can be defined without the use of localizations. On this occasion, we are also interested in groups of internal symmetries. They are generated by special vector fields, the infinitesimal symmetries.

In the actual literature, differential equations are as a rule considered in finite-dimensional jet spaces. Then the internal and external symmetries become rather delicate and differ from our concepts since the higher-order symmetries are not taken into account. We will not discuss such conceptual confusion in this paper with the belief that the following two remarks (and Remark 5) should be quite sufficient in this respect.

Remark 1 (on the symmetries). The true structure of the jet space is determined by the contact module   which involves all contact forms Then the above morphisms   given in ((1), (6)) are characterized by the property . Recall that invertible morphisms are automorphisms. Let us introduce the subspace of all points (9). This is equipped with the restriction of the contact module. Recall that we are interested only in the case (abbreviation of ). Let be the restriction of . If is a morphism then is a morphism in the sense that . Recall that we have the internal symmetry, if is moreover invertible. If also is invertible, we have the external symmetry  . The internal symmetries can be defined without any reference to and as follows. Let be any invertible mapping such that . This can be always extended to a morphism   of the ambient jet space. (Hint, recurrence (6) holds true both in and in .) So we may conclude that such is just the internal symmetry. Moreover, if there exists invertible extension of , then is even the external symmetry but the latter concept already depends on the localization of in .

Remark 2 (on infinitesimal symmetries). Let us consider a vector field on the jet space . Let us moreover suppose from now on (where denotes the Lie derivative see also Definition 8). In common terminology, such vector fields are called generalized (higher-order, Lie-Bäcklund) infinitesimal symmetries of the jet space . However need not in general generate any true group of transformations and we therefore prefer the “unorthodox” term a variation   here. (See Section 7 and especially Remark 35 where the reasons for this term are clarified.) The common term infinitesimal symmetry is retained only for the favourable case when generates a local one-parameter Lie group [5]. Let us consider the above subspace . If is tangent to , then there exists the natural restriction of to . Clearly and we speak of the (internal) variation  . If moreover generates a group in , we have the (internal) infinitesimal symmetry  . The internal concepts on can be easily introduced without any reference to the ambient space . This is not the case for the concept of the external infinitesimal symmetry   which supposes that appropriate extension   of on the ambient space   generates a Lie group.

We deal only with the internal symmetries and infinitesimal symmetries in this paper. It is to be noted once more that infinite-dimensional underlying spaces are necessary if we wish to obtain a coherent theory. The common technical tools invented in the finite-dimensional spaces will be only slightly adapted; alas, the ingenious methods proposed, for example, in [68] seem to be not suitable for this aim and so we undertake the elementary approach [9] here.

2. Technical Tools

We introduce infinite-dimensional manifold modelled on the space with local coordinates in full accordance with [9]. The manifold is equipped with the structural algebra   of -smooth functions expressed as in terms of coordinates. Transformations (mappings) are (locally) given by certain formulae and analogous (invertible) formulae describe the change of coordinates at the overlapping coordinate systems.

Let be the -module of differential 1-forms The familiar rules of exterior calculus can be applied without any change, in particular for the above transformation .

Let be the -module of vector fields . In terms of coordinates we have where the coefficients may be quite arbitrary. We identify with the linear functional on determined by the familiar duality pairing With this principle in mind, if certain forms generate the -module, then the values uniquely determine the vector field and (17) can be very expressively (and unorthodoxly) recorded by This is a mere symbolical record, not the true infinite series. However, if is a basis of the module in the sense that every admits a unique representation , finite sum) then the coefficients can be quite arbitrary and (18) may be regarded as a true infinite series. The arising vector fields provide a weak basis (infinite expansions, see [9]) of dual to the basis of . In this transcription, (15) is alternatively expressed as We recall the Lie derivative   acting on exterior differential forms. The image of a vector field defined by the property need not exist. It is defined if is invertible.

We consider various submodules of differential forms together with the relevant orthogonal submodules consisting of all vector fields such that . The existence of (local) -bases in all submodules of to appear in our reasonings is tacitly postulated. Dimension of an -module is the number of elements of an -basis. Omitting some “exceptional points,” it may be confused with the dimension of the corresponding -module (the localization) at a fixed place . On this occasion, it should be noted that the image of a tangent vector at exists as a vector at the place .

Let us also remark with regret that any rigorous exposition of classical analysis in the infinite-dimensional space is not yet available; however, certain adjustments of finite-dimensional results are not difficult. For instance, the following invertibility theorem will latently occur in the proof of Theorem 20.

Theorem 3. A mapping is invertible if and only if any of the following equivalent conditions is satisfied: the pull-back is invertible, the pull-back is invertible, and if is a (fixed, equivalently: arbitrary) basis of module , then again is a basis.

Hint. A nonlinear version of the familiar Gauss elimination procedure for infinite dimension [9] provides a direct proof with difficulties concerning the definition domain of the resulting inverse mapping. Nevertheless if is moreover a morphism of a diffiety (see Definition 8) then the prolongation procedure ensures the local existence of in the common sense.

3. Fundamental Concepts

We introduce a somewhat unusual intrinsical approach to underdetermined systems of ordinary differential equations in terms of the above underlying space , a submodule of differential 1-forms, and its orthogonal submodule of vector fields.

Definition 4. A codimension one submodule is called a diffiety if there exists a good filtration by finite-dimensional submodules such that To every subset , let denote the submodule with generators . Since (easy), the second requirement (23) can be a little formally simplified as .

Remark 5. This is a global coordinate-free definition; however, we again deal only with the local theory from now on in the sense that the definition domains (of filtrations (22), of independent variable to follow, and so on) are not specified. It should be noted on this occasion that the common geometrical approach [68] to differential equations rests on the use of the rigid structure of finite-order jets. Many classical concepts then become incorrect, if the higher-order mappings are allowed but we cannot adequately discuss this important topic here. Rather subtle difficulties are also passed over already in the common approach to the fundamental jet theory. For instance, smooth curves in the plane with coordinates are parametrized either by (i.e., ) or by coordinate (i.e., ) in the common so-called “geometrical” approach [68]. However, then already the Lie’s classical achievements concerning contact transformations [10, 11] with curves parametrized either by or by cannot be involved. Quite analogously, the “higher-order” parameterizations and mappings [25] are in fact rejected in the common “rigid” jet theory with a mere point symmetries.

Definition 6. Let a differential generate together with the total module of all differential 1-forms. Then is called the independent variable to diffiety . The vector field (abbreviation) such that is called total (or formal) derivative of with respect to the independent variable . This vector field is a basis of the one-dimensional module for every fixed particular choice of the independent variable .

Remark 7. Let us state some simple properties of diffieties. The proofs are quite easy and may be omitted. A form is lying in if and only if . In particular in accordance with the identities (This trivial property clarifies the more restrictive condition (23).) Moreover clearly and in particular for all coordinates. We have very useful -generators of diffiety . The independent variable and the filtrations (22) can be capriciously modified. In particular the -lift [9] with large enough ensures that for all . We will be, however, interested just in the reverse concept “” latently involved in the “standard adaptation” of filtrations to appear later on.

Definition 8. A transformation is called a morphism of the diffiety if . Invertible morphisms are automorphisms (or symmetries) of . A vector field satisfying is called the variation of . If moreover (locally) generates a one-parameter group of transformations, we speak of the infinitesimal symmetry   of diffiety .

Remark 9. Let us mention the transformation groups in more detail. A local one-parameter group of transformations is given by certain formulae in terms of local coordinates, where is supposed. Then the special vector field (15) defined by is called the infinitesimal transformation of the group (29). In the opposite direction, we recall that a general vector field (15) generates the local group (29) if and only if the Lie system is satisfied. Alas, a given vector field (19) need not in general generate any transformation group since the Lie system need not admit any solution (29).

With all fundamental concepts available, let us eventually recall the familiar and thoroughly discussed in [9] interrelation between the diffieties and the corresponding classical concept of differential equations for the convenience of reader. In brief terms, the idea is quite simple. A given system of differential equations is represented by a system of Pfaffian equations and the module generated by such 1-forms is just the diffiety. More precisely, we deal with the infinite prolongations as follows.

In one direction, let a system of underdetermined ordinary differential equations be given. We may deal with the first-order system without any true loss of generality. Then (32) completed with provides the infinite prolongation. The corresponding diffiety is generated by the forms in the space with coordinates Clearly is the total derivative and the submodules of all forms (34) with determine a quite simple filtration (22) with respect to the order of contact forms. (Hint: use the formulae and .) However, there exist many other and more useful filtrations; see the examples to follow later on.

The particular case of the empty system (32) can be easily related to the case of the jet space of all -parametrized curves in of the Section 1. The relevant diffiety is identified with the module of all contact forms (11), of course.

In the reverse direction, let a diffiety be given on the space . In accordance with (27), the forms generate . So we have the Pfaffian system and therefore the system of differential equations of rather unpleasant kind. Then, due to the existence of a filtration (22) and (23), one can obtain also the above classical system of differential equations (32) together with the prolongation (33) by means of appropriate change of coordinates [9]. This is, however, a lengthy procedure and a shorter approach can be described as follows. Let the second requirement (23) be satisfied, if . Suppose that the forms generate module . Then all forms generate the diffiety . The corresponding Pfaffian system is equivalent to certain infinite prolongation of differential equations, namely, (direct verification), and in general We have the infinite prolongation of the classical system and this is just the system that corresponds to diffiety .

Altogether taken, differential equations uniquely determine the corresponding diffieties; however, a given diffiety leads to many rather dissimilar but equivalent systems of differential equations with regard to the additional choice of dependent and independent variables.

Remark 10. Definitions 48 make good sense even if is a finite-dimensional manifold and then provide the well-known intrinsical approach to determined systems of differential equations. They are identified with vector fields (better, fields of directions) in the finite-dimensional space . Choosing a certain independent variable , the equations are represented by the vector field or, more visually, by the corresponding -flow. The general theory becomes trivial; we may, for example, choose for all in filtration (22).

4. On the Structure of Diffieties

Definition 11. To every submodule of a diffiety , let be the submodule of all such that . Filtration (22) and (23) is called a standard one, if For every , the first condition ensures that the inclusions are equivalent and the second condition ensures that implies .

Theorem 12. Appropriate adaptation of some lower-order terms of a given filtration (22) and (23) provides a standard filtration in a unique manner [9]. Equivalently and in more detail, there exists unique standard filtration such that for appropriate and all large enough. Equivalently and briefly, there exists unique standard filtration such that for appropriate .

Proof. The mapping naturally induces certain -homomorphism of factor modules denoted by for better clarity. Homomorphisms are surjective and therefore even bijective for all large enough, say for . However, the injectivity of implies . It follows that we have strongly decreasing sequence which necessarily terminates with the stationarity . Denoting we have the sought strongly increasing standard filtration of diffiety . In particular .

Proof of Theorem 12 was of the algorithmical nature and provides a useful standard basis of diffiety as follows. Assume that the forms (recall that whence ) and moreover the classes of forms (recall that is injective mapping), the classes of forms (recall that is injective mapping), and in general the classes of forms Alternatively saying, the following forms constitute a basis: and so on. Let us denote In terms of this notation we have the standard basis of .

Clearly and it follows that there is only a finite number of initial forms with the lower zero indice. The following forms satisfy the recurrence and the (equivalent) congruence In this sense, the linearly independent forms    are generalizations of the classical contact forms    of the jet theory.

Theorem 13. Let be a standard filtration of diffiety . Then the submodule is generated by all differentials .

Proof. First assume . Then whence . Clearly for appropriate . This implies , if therefore . It follows that contains all differentials .
Conversely let . Due to the equality , we have whence ), consequently It follows that identically by using and (55). (Hint: look at assumed top order product where . Then involves only one summand with which is impossible since .) Therefore and the Frobenius theorem can be applied. Module has a basis consisting of total differentials.

Definition 14. We may denote since this module does not depend on the choice of the filtration (22). Together with the original basis occurring in (53), there exists alternative basis with differentials. In the particular case , hence, , we speak of a controllable diffiety .

Remark 15. The controllability is a familiar concept of the theory of underdetermined ordinary differential equations or Pfaffian systems in finite-dimensional spaces [12]; however, some aspects due to diffieties are worth mentioning here. If is a nontrivial module, the underlying space is fibered by the leaves depending on parameters. A curve is called a solution of diffiety , if . Since , we have therefore every solution of diffiety is contained in a certain leaf (the Figure 2(a)).

In the controllable case, such foliation of the space does not exist. However, the construction of the standard filtration need not be of the “universal nature.” There may exist some “exceptional points” where the terms of the standard basis are not independent. We may even obtain a solution consisting of such exceptional points and then there appears the “infinitesimal leaf” of the noncontrollability along which means that is a Mayer extremal (the Figure 2(b)). We refer to article [13] inspired by the beautiful paper [14]. In the present paper, such exceptional points are tacitly excluded. They produce singularities of the symmetry groups and deserve a special, not yet available approach. It should be noted that the noncontrollable case also causes some technical difficulties. We may however suppose without much loss of generality since the noncontrollable diffiety can be restricted to a leaf and regarded as a diffiety depending on parameters .

Theorem 16. The total number of initial forms does not depend on the choice of the good filtration (22).

Proof. Filtration (22) differs from the standard filtration only in lower terms whence Let another filtration of diffiety provide (corresponding standard filtration and therefore) certain number of (other) initial forms. Then However for appropriate and whence by using (23) and the equality easily follows.

5. On the Morphisms and Variations

A huge literature on the point symmetries (scheme (a) of Figure 3, the order of derivatives is preserved) of differential equations is available. On the contrary, we can mention only a few fundamental principles for the generalized (or higher-order) symmetries (scheme (c) of Figure 3) since the general theory deserves quite another paper. Our modest aim is to clarify a little the mechanisms of the particular examples to follow. We will also deal with generalized (or higher-order) groups of symmetries and the relevant generalized infinitesimal symmetries (scheme (b) Figure 3) with ambiguous higher-order invariant subspaces (the dotted lines). Figure 3 should be therefore regarded as a rough description of the topics to follow and we also refer to Section 9 for more transparent details. The main difficulty of the higher-order theory lies in the fact that the dotted domains are not known in advance. Modules represent the “natural” filtration with respect to the primary order of contact forms in the ambient jet space, see the examples. They depend on the accidental inclusion mentioned in Section 1 and do not have any true geometrical sense in the internal approach. It is to be therefore surprisingly observed that the seemingly “exotic” at the first glance concept of higher-order transformations of Section 1 should be regarded for reasonable and the only possible in the coordinate free theory. On the other hand, an important distinction between the group-like morphisms with large number of finite-dimensional invariant subspaces (scheme (a) and (b)) and the genuine order-destroying morphisms without such subspaces (scheme (c)) is of the highest importance.

We are passing to rigorous exposition. Let us recall the diffiety on the space , the independent variable with the corresponding vector field , the controllability submodule with the basis , and a standard basis of diffiety .

Let us begin with morphisms.

Lemma 17. If is a morphism of then and the recurrence modulo holds true.

Proof. If is a morphism then therefore (use Theorem 13) and . It follows that modulo and . This implies (61) by comparing both factors of .

Remark 18. On this occasion, the following useful principles of calculation are worth mentioning: and in general In terms of notation (21), we conclude that and therefore if the morphism of diffiety is invertible.

Let us turn to invertible morphisms.

Lemma 19. The inverse of a morphism again is a morphism.

Proof. Assume ). Then where . Hence and therefore .

We have if is a morphism and moreover hence in the invertible case. The converse and rather useful assertion is as follows.

Theorem 20. A morphism of diffiety is invertible if and only if .

This may be obtained easily from the following result.

Lemma 21. Let and . Then is invertible.

Proof. Proof of the Lemma 21 is analogous as in [2, Theorem 2] and we briefly recall only the main principles here. It is sufficient to prove the invertibility of .
Assuming then by virtue of recurrence (61). It follows that and is surjective. We prove that   is even injectivity by using the well-known algebraical interrelation between filtrations and gradations.
Let us introduce filtrations (, resp.) as follows: the submodule is generated by and all forms where . We also introduce the gradations (formally ). It follows that the naturally induced mapping is surjective and it is sufficient to prove that this induced is also injective.
We are passing to the most delicate part of the proof. The surjectivity of implies that for large enough. Therefore by applying the recursion (61) which implies On the other hand, assume the noninjectivity therefore the existence of a nontrivial identity Then by applying operator and recurrence (61). Due to the existence of such identities, it follows that and this is a contradiction.

Remark 22. Recall that if is a mapping and a submodule, then denotes the submodule with generators in accordance with the common practice in the algebraical module theory. Let in particular be a diffiety and assume for simplicity. Then module is generated by all forms and therefore by all forms , see Lemma 17. It follows that the invertibility of the morphism depends only on the properties of the forms , see Lemma 21. In this sense, the invertibility problem is reduced to the finite-dimensional reasonings.

We turn to the variations.

Lemma 23. A vector field is a variation of diffiety if and only if and all are functions only of variables .

Proof. We suppose which is equivalent to the congruences by using ((26) and (55)). So we have obtained (72) and moreover identities .
It is sufficient to prove that the latter identities imply . However, every differential can be represented as in terms of the standard basis. Assuming in particular , we have already obtained the equation and then identities easily follow by applying the common rule together with (26). This concludes the proof.

Theorem 24. A variation of diffiety is infinitesimal symmetry of if and only if all forms are contained in a finite-dimensional module.

We omit lengthy proof and refer to more general results [5, Lemma 5.4, Theorem 5.6, and especially Theorem 11.1]. In future examples, we apply other and quite elementary arguments in order to avoid the nontrivial Theorem 24.

Remark 25. It follows from Lemma 23 that variations of diffiety can be represented by the universal series where are arbitrary composed functions and are arbitrary functions in . We have explicit formulae for all variations (in common terms, for all Lie-Bäcklund infinitesimal symmetries) of a given system of ordinary differential equations. Recall that these variations need not generate any true group, and though the criterion in Theorem 24 is formally simple, it is not easy to be applied. Lemma 17 can be regarded as a counterpart to Lemma 23 since it ensures quite analogous result for the morphism or, better saying, for the pullback of a morphism. In more detail, the quite arbitrary choice of the initial terms of recurrence (61) is in principle possible but provides a mere formal result (corresponding to the formal nature of variations ) and does not ensure the existence of true morphism . We may refer to articles [2, 3] where the formal part (the algebra) is distinguished from the nonformal part (the analysis) in the higher-order algorithms.

We conclude this Section with the only gratifying result [9, point on page 40].

Theorem 26. The standard filtration is unique in the case .

Proof. Let us take a fixed filtration (22) and the corresponding standard filtration (46). Since , we have only one initial form and therefore is a basis of ; see (53). Let us take another standard filtration . Then the module has certain basis These forms together with all generate the module and this is possible only if . We conclude that which implies hence for all .

Remark 27. It follows that in the particular case , every symmetry and infinitesimal symmetry preserves all terms of the (unique) standard filtration. So we have a large family of finite-dimensional subspaces of the underlying space which are preserved too. The classical methods acting in finite-dimensional spaces uniquely determined in advance can be applied and are quite sufficient in this case  .

Remark 28. In more generality, one could also consider two diffieties and on the underlying spaces and , respectively. Though we do not deal with the isomorphism problems of two diffieties and here, let us mention that such isomorphism is defined as invertible mapping of underlying spaces satisfying . Quite equivalent “absolute equivalence” problem was introduced in [15] and resolved just for the case (in our terminology) by using finite-dimensional methods. We have discovered alternative approach here: the isomorphism identifies the unique standard filtrations of and of . On this occasion, it is worth mentioning Cartan’s pessimistic notice (rather unusual in his work) to the case “Je dois ajourter que la géneralization de la théorie de l’equivalence absolu aux systémes differentiels dont la solution générale dépend de deux functions arbitraires d’un argument n’est pas immédiate et souléve d’asses grosses difficultiés.” The same notice can be literally repeated also for the theory of the higher-order symmetries treated in this paper.

6. The Order-Preserving Case of Infinitesimal Symmetries

We are passing to the first example which intentionally concerns the well-known “towering” problem in order to examine our method reliably. Let us deal with infinitesimal symmetries of differential equation involving two unknown functions and . In external theory, (77) is identified with the subspace defined by the conditions in the jet space . We use simplified notation of coordinates and contact forms here. We are, however, interested in internal theory, that is, in the diffiety corresponding to (77). Diffiety appears if the contact forms are restricted to the subspace . In accordance with the common practice, let us again simplify as the notation of the restrictions to and moreover will be regarded as a vector field on from now on.

Let us outline the lengthy path of future reasonings for the convenience of reader. We begin with preparatory points (ι)–(ιιι). The underlying space together with the diffiety is introduced and the standard basis , abbreviation of diffiety is determined. The standard basis is related to the “common” basis of by means of formulae (93). We obtain explicit representation (99) for the variations   with two arbitrary functions   and as the final result. Variations generating the true group (i.e., the infinitesimal symmetries   of ) satisfy certain strong conditions discovered in points and . The conditions are expressed by the resolving system (107) and (108) or, alternatively, by (112)–(114) only in terms of the functions , , , and . This rather complicated resolving system which does not provide any clear insight is equivalent to much simpler crucial requirements (121) or (125) on the actual structure of function ; see the central points (νι)–(νιιι). Then the subsequent points are devoted to the explicit solution of these equations (125). This is a mere technical task of traditional mathematical analysis and we omit comments at this place.

(ι) The diffiety. Let us introduce space equipped with coordinates . Then are merely composed functions. The forms provide a basis of the diffiety ; however, all forms are also lying in as follows from the obvious rule: and the inclusion .

(ιι) Standard Filtration. There exists the “natural” filtration of diffiety with respect to the order: submodule involves the forms with . Alternatively saying, is a basis of and Clearly if as follows from (84). However, (Figure 4(a)) therefore Then may be taken for a basis of module (Figure 4(b)).

Moreover hence constitute a basis of module (Figure 4(c)) and finally Therefore assuming from now on, the form may be taken for a basis of module . We have obtained the standard filtration where forms provide a basis of module .

Abbreviating from now on, explicit formulae where and can be easily found. They will be sufficient in calculations to follow. Recall that we suppose that the inequality (90) hold true, hence .

(ιιι) Variations. We deal with vector fields (the notation (75) with indices is retained) on the space . Recall that is a variation if . In terms of coordinates, the conditions are where the first and third equations are merely recurrences while the middle equation causes serious difficulties (a classical result. Hint: use ). By using the alternative formula the conditions slightly simplify (Hint: apply the rule to the forms However, by virtue of Lemma 23 and standard filtration, we have explicit formula for the variations where and are arbitrary functions. One can then easily obtain explicit formulae for all coefficients in (97) and in (95) by using the left-hand identities (93). They need not be stated here.

(ιν) Infinitesimal Transformations. We refer to Remark 27: variation   is infinitesimal symmetry if and only if for appropriate multiplier . In explicit terms, we recall formula where and therefore clearly So denoting requirement (100) reads where and () should be moreover inserted. It follows that requirement (100) is equivalent to the so-called resolving system Moreover