Department of Theoretical Physics,A. Razmadze Institute of Mathematics,1 Aleksidze Street, Tbilisi 0193, Georgia

Symmetries play essential role in dynamical systems, because they usually simplifyanalysis of evolution equations and often provide quite elegant solution of problems that otherwise wouldbe difficult to handle. In Lagrangian and Hamiltonian dynamical systems special role is playedby Noether symmetries — an important class of symmetries that leave action invariantand have some exceptional features. In particular, Noether symmetries deservedspecial attention due to celebrated Noether's theorem, that established correspondencebetween symmetries, that leave action functional invariant, and conservation lawsof Euler-Lagrange equations. This correspondence can be extended to Hamiltoniansystems where it becomes more tight and evident then in Lagrangian case and gives riseto Lie algebra homomorphism between Lie algebra of Noether symmetries and algebra ofconservation laws (that form Lie algebra under Poisson bracket).

Role of symmetries that are not of Noether type has been suppressed for quite a long time.However, after some publications of Hojman, Harleston, Lutzky and others(see [16], [36], [39],[40], [49]-[57])it became clear that non-Noether symmetries also can play important role inLagrangian and Hamiltonian dynamics. In particular, according to Lutzky[51], in Lagrangian dynamics there is definite correspondence between non-Noether symmetries andconservation laws. Moreover, each generator of non-Noether symmetrymay produce whole family of conservation laws (maximal number of conservation laws that canbe associated with non-Noether symmetry via Lutzky's theorem is equal to the dimension ofconfiguration space of Lagrangian system). This fact makes non-Noether symmetries especiallyvaluable in infinite dimensional dynamical systems, where potentially one can recoverinfinite sequence of conservation laws knowing single generator of non-Noether symmetry.

Existence of correspondence between non-Noether symmetries and conserved quantitiesraised many questions concerning relationship among this type of symmetries andother geometric structures emerging in theory of integrable models.In particular one could notice suspicious similarity between the method of constructingconservation laws from generator of non-Noether symmetry andthe way conserved quantities are produced in either Lax theory, bi-Hamiltonian formalism,bicomplex approach or Lenard scheme.It also raised natural question whether set of conservation laws associated with non-Noethersymmetry is involutive or not, and since it appeared that in general it may not be involutive,there emerged the need of involutivity criteria, similar to Yang-Baxter equation used in Lax theoryor compatibility condition in bi-Hamiltonian formalism and bicomplex approach.It was also unclear how to construct conservation laws in case of infinite dimensionaldynamical systems where volume forms used in Lutzky's construction are no longer well defined.Some of these questions were addressed in papers [11]-[14],while in the present review we would like to summarize all these issues and to provide someexamples of integrable models that possess non-Noether symmetries.

Review is organized as follows. In first section we briefly recall some aspects of geometricformulation of Hamiltonian dynamics. Further, in second section, correspondencebetween non-Noether symmetries and integrals of motion in regular Hamiltonian systems isdiscussed. Lutzky's theorem is reformulated in terms of bivector fieldsand alternative derivation of conserved quantities suitable for computations in infinitedimensional Hamiltonian dynamical systems is suggested. Non-Noether symmetries oftwo and three particle Toda chains are used to illustrate general theory.In the subsequent section geometric formulation of Hojman's theorem [36]is revisited and some examples are provided. Section 4 reveals correspondence betweennon-Noether symmetries and Lax pairs. It is shown that non-Noether symmetry canonicallygives rise to a Lax pair of certain type. Lax pair is explicitly constructed in termsof Poisson bivector field and generator of symmetry. Examples of Toda chains are discussed.Next section deals with integrability issues. An analogue of Yang-Baxter equationthat, being satisfied by generator of symmetry, ensures involutivity of setof conservation laws produced by this symmetry, is introduced.Relationship between non-Noether symmetries and bi-Hamiltonian systemsis considered in section 6. It is proved that under certain conditions,non-Noether symmetry endows phase space of regular Hamiltonian system withbi-Hamiltonian structure. We also discuss conditions under which non-Noethersymmetry can be "recovered" from bi-Hamiltonian structure.Theory is illustrated by example of Toda chains. Next section is devoted tobicomplexes and their relationship with non-Noether symmetries. Special kindof deformation of De Rham complex induced by symmetry is constructed in terms ofPoisson bivector field and generator of symmetry.Samples of two and three particle Toda chain are discussed.Section 8 deals with Frölicher-Nijenhuis recursion operators.It is shown that under certain condition non-Noether symmetrygives rise to invariant Frölicher-Nijenhuis operator on tangentbundle over phase space.The last section of theoretical part contains some remarks on action of one-parametergroup of symmetry on algebra of integrals of motion. Special attention is devoted toinvolutivity of group orbits.

Subsequent sections of present review provide examples of integrable modelsthat possess interesting non-Noether symmetries. In particular section 10 revealsnon-Noether symmetry of $n$-particle Toda chain. Bi-Hamiltonian structure,conservation laws, bicomplex, Lax pair and Frölicher-Nijenhuis recursionoperator of Toda hierarchy are constructed using this symmetry.Further we focus on infinite dimensional integrable Hamiltonian systems emergingin mathematical physics. In section 11 case of Korteweg-de Vriesequation is discussed. Symmetry of this equation is identified and used in constructionof infinite sequence of conservation laws and bi-Hamiltonian structure ofKdV hierarchy. Next sectionis devoted to non-Noether symmetries of integrable systems of nonlinear water wave equations,such as dispersive water wave system, Broer-Kaup system and dispersiveless long wave system.Last section focuses on Benney system and its non-Noether symmetry, that appears to be local,gives rise to infinite sequence of conserved densities of Benney hierarchy and endows it withbi-Hamiltonian structure.

The basic concept in geometric formulation of Hamiltonian dynamicsis notion of symplectic manifold. Such a manifold plays the role ofthe phase space of the dynamical system and therefore many propertiesof the dynamical system can be quite effectively investigated in the frameworkof symplectic geometry. Before we consider symmetries of the Hamiltonian dynamicalsystems, let us briefly recall some basic notions from symplectic geometry.

The symplectic manifold is a pair $(M,\; \omega )$where $M$ is smooth even dimensional manifold and $\omega $is a closeddω = 0 and nondegenerate 2-form on $M$. Being nondegenerate means thatcontraction of arbitrary non-zero vector field with $\omega $ does not vanishi_{X}ω = 0 ⇔ X = 0 (here $i$_{X} denotes contraction of the vector field $X$with differential form). Otherwise one can say that $\omega $is nondegenerate if its n-th outer power does not vanish($\omega n\ne \; 0$) anywhere on $M$.In Hamiltonian dynamics $M$ is usually phase space of classical dynamical systemwith finite numbers of degrees of freedom and the symplectic form $\omega $is basic object that defines Poisson bracket structure, algebra of Hamiltonian vector fieldsand the form of Hamilton's equations.

The symplectic form $\omega $ naturally defines isomorphism between vector fieldsand differential 1-forms on $M$ (in other words tangent bundle $TM$of symplectic manifold can be quite naturally identified withcotangent bundle $T*M$).The isomorphic map $\Phi $_{ω} from $TM$ into$T*M$ is obtained by taking contractionof the vector field with $\omega $Φ_{ω}: X → − i_{X}ω (minus sign is the matter of convention). This isomorphism gives rise to natural classificationof vector fields. Namely, vector field $X$_{h} is said to be Hamiltonianif its image is exact 1-form or in other words if it satisfies Hamilton's equationi_{Xh}ω + dh = 0 for some function $h$ on $M$.Similarly, vector field $X$ is called locally Hamiltonian if it's image is closed 1-formi_{X}ω + u = 0, du = 0

One of the nice features of locally Hamiltonian vector fields, known as Liouville's theorem,is that these vector fields preserve symplectic form $\omega $.In other words Lie derivative of the symplectic form $\omega $along arbitrary locally Hamiltonian vector field vanishesL_{X}ω = 0 ⇔ i_{X}ω + du = 0, du = 0 Indeed, using Cartan's formula that expresses Lie derivative in terms of contraction andexterior derivativeL_{X} = i_{X}d + di_{X} one getsL_{X}ω = i_{X}dω + di_{X}ω =di_{X}ω (since $d\omega \; =\; 0$) but according to the definition of locally Hamiltonianvector fielddi_{X}ω = − du = 0 So locally Hamiltonian vector fields preserve $\omega $ and vise versa,if vector field preserves symplectic form $\omega $ then it is locally Hamiltonian.

Clearly, Hamiltonian vector fields constitute subset of locally Hamiltonian ones sinceevery exact 1-form is also closed. Moreover one can notice that Hamiltonian vector fields formideal in algebra of locally Hamiltonian vector fields. This fact can be observed as follows.First of all for arbitrary couple of locally Hamiltonian vector fields $X,\; Y$we have $L$_{X}ω = L_{Y}ω = 0 andL_{X}L_{Y}ω − L_{Y}L_{X}ω= L_{[X , Y]}ω = 0 so locally Hamiltonian vector fields form Lie algebra (corresponding Lie bracket is ordinarycommutator of vector fields). Further it is clear that for arbitrary Hamiltonian vector field$X$_{h} and locally Hamiltonian one $Z$ one hasL_{Z}ω = 0 andi_{Xh}ω + dh = 0 that impliesL_{Z}(i_{Xh}ω + dh) = L_{[Z , Xh]}ω + i_{Xh}L_{Z}ω +dL_{Z}h= L_{[Z , Xh]}ω + dL_{Z}h = 0 thus commutator $[Z\; ,\; X$_{h}] is Hamiltonian vector field$X$_{LZh},or in other words Hamiltonian vector fields form ideal in algebra of locallyHamiltonian vector fields.

Isomorphism $\Phi $_{ω} can be extended tohigher order vector fields and differential forms by linearity and multiplicativity.Namely,Φ_{ω}(X ∧ Y) =Φ_{ω}(X) ∧ Φ_{ω}(Y) Since $\Phi $_{ω} is isomorphism, the symplectic form $\omega $has unique counter image $W$ known as Poisson bivector field.Property $d\omega \; =\; 0$ together with non degeneracy implies that bivectorfield $W$ is also nondegenerate ($Wn\ne \; 0$) and satisfiescondition[W , W] = 0 where bracket $[\; ,\; ]$ known as Schouten bracket or supercommutator, is actuallygraded extension of ordinary commutator of vector fields to the case of multivector fields,and can be defined by linearity and derivation property[C_{1} ∧ C_{2} ∧ ... ∧ C_{n} ,S_{1} ∧ S_{2} ∧ ... ∧ S_{n}] = (− 1)^{p + q}[C_{p} , S_{q}] ∧C_{1} ∧ C_{2} ∧ ... ∧ Ĉ_{p} ∧ ... ∧ C_{n} ∧ S_{1} ∧ S_{2} ∧ ... ∧ Ŝ_{q} ∧ ...∧ S_{n} where over hat denotes omission of corresponding vector field.In terms of the bivector field $W$ Liouville's theorem mentioned above can berewritten as follows[W(u) , W] = 0 ⇔ du = 0 for each 1-form $u$. It follows from graded Jacoby identity satisfied by Schoutenbracket and property $[W\; ,\; W]\; =\; 0$ satisfied by Poisson bivector field.

Being counter image of symplectic form, $W$ gives rise to map$\Phi $_{W}, transforming differential 1-forms into vector fields,which is inverted to the map $\Phi $_{ω} and is defined byΦ_{W}: u → W(u); Φ_{W}Φ_{ω} = id Further we will often use these maps.

In Hamiltonian dynamical systems Poisson bivector field is geometric object thatunderlies definition of Poisson bracket — kind of Lie bracket on algebra ofsmooth real functions on phase space. In terms of bivector field $W$Poisson bracket is defined by{f , g} = W(df ∧ dg) The condition $[W\; ,\; W]\; =\; 0$ satisfied by bivector field ensures thatfor every triple $(f,\; g,\; h)$ of smoothfunctions on the phase space the Jacobi identity{f{g , h}} + {h{f , g}} + {g{h , f}} = 0. is satisfied.Interesting property of the Poisson bracket is that map from algebra of real smooth functionson phase space into algebra of Hamiltonian vector fields, defined by Poisson bivector fieldf → X_{f} = W(df) appears to be homomorphism of Lie algebras. In other words commutator of two vector fieldsassociated with two arbitrary functions reproduces vector field associated with Poissonbracket of these functions[X_{f} , X_{g}] = X_{{f , g}} This property is consequence of the Liouville theorem and definition of Poisson bracket.Further we also need another useful property of Hamiltonian vector fields and Poisson bracket{f , g} = W(df ∧ dg) = ω(X_{f} ∧ X_{g}) =L_{Xf}g = − L_{Xg}g it also follows from Liouville theoremand definition of Hamiltonian vector fields and Poisson brackets.

To define dynamics on $M$ one has to specify time evolution of observables(smooth functions on $M$). In Hamiltonian dynamical systems time evolutionis governed by Hamilton's equationd dt f = {h , f} where $h$ is some fixed smooth function on the phase space called Hamiltonian.In local coordinate frame $z$_{b} bivector field $W$has the formW = W_{bc} ∂ ∂z_{b} ∧ ∂ ∂z_{c} and the Hamilton's equation rewritten in terms of local coordinates takes the formż_{b} = W_{bc}∂h ∂z_{b}

Now let us focus on symmetries of Hamilton's equation (24).Generally speaking, symmetries play very important role in Hamiltonian dynamicsdue to different reasons. They not only give rise to conservation laws butalso often provide very effective solutions to problems that otherwise would be difficultto solve. Here we consider special class of symmetries of Hamilton's equationcalled non-Noether symmetries. Such a symmetries appear to be closely related tomany geometric concepts used in Hamiltonian dynamics including bi-Hamiltonian structures,Frölicher-Nijenhuis operators, Lax pairs and bicomplexes.

Before we proceedlet us recall that each vector field $E$ on the phase space generatesthe one-parameter continuous group of transformationsg_{z} = e^{zLE} (here $L$ denotes Lie derivative)that acts on the observables as followsg_{z}(f) = e^{zLE}(f) =f + zL_{E}f + ½(zL_{E})^{2}f + ⋯ Such a group of transformation is called symmetry of Hamilton's equation (24)if it commutes with time evolution operatord dt g_{z}(f)= g_{z}(d dt f) in terms of the vector fields this condition means that the generator$E$ of the group $g$_{z} commutes with the vector field$W(h)\; =\; \{h\; ,\; \}$, i. e.[E , W(h)] = 0. However we would like to consider more generalcase where $E$ is time dependent vector field on phase space. In this case(30) should be replaced with∂ ∂t E = [E , W(h)].

Further one should distinguish between groups of symmetry transformations generated by Hamiltonian,locally Hamiltonian and non-Hamiltonian vector fields. First kind of symmetriesare known as Noether symmetries and are widely used in Hamiltonian dynamics due to theirtight connection with conservation laws. Second group of symmetries is rarely used. While third group of symmetries that further will be referredas non-Noether symmetries seems to play important role in integrability issues due totheir remarkable relationship with bi-Hamiltonian structures andFrölicher-Nijenhuis operators. Thus if in addition to (30) thevector field $E$ does not preserve Poisson bivector field $[E\; ,\; W]\; \ne \; 0$ then$g$_{z} is called non-Noether symmetry.

Now let us focus on non-Noether symmetries. We would like to show that the presence ofsuch a symmetry essentially enriches the geometry of the phase spaceand under the certain conditions can ensure integrability of the dynamical system.Before we proceed let us recall that the non-Noether symmetry leads to a number ofintegrals of motion. More precisely therelationship between non-Noether symmetries and the conservation laws is described bythe following theorem. This theorem was proposed by Lutzky in [51].Here it is reformulated in terms of Poisson bivector field.

Besides Hamiltonian dynamical systems that admit invariant symplectic form$\omega $, there are dynamical systems that either are not Hamiltonian oradmit Hamiltonian realization but explicit form of symplectic structure $\omega $is unknown or too complex. However usually such a dynamical systems possess invariant volume form$\Omega $ which like symplectic form can be effectively used in construction ofconservation laws. Note that volume form for given manifold is arbitrary differential formof maximal degree (equal to the dimension of manifold).In case of regular Hamiltonian systems, n-th outer power of the symplectic form $\omega $naturally gives rise to the invariant volume form known as Liouville formΩ = ω^{n} and sometimes it is easier to work with $\Omega $ rather then with symplectic form itself.In generic Liouville dynamical system time evolution is governed by equations of motiond dt f = X(f) where $X$ is some smooth vector field that preserves Liouville volume form$\Omega $d dt Ω = L_{X}Ω = 0 Symmetry of equations of motion still can be defined by conditiond dt g_{z}(f)= g_{z}(d dt f) that in terms of vector fields implies that generator of symmetry $E$ shouldcommute with time evolution operator $X$[E , X] = 0 Throughout this chapter symmetry will be called non-Liouville if it is not conformal symmetryof $\Omega $, or in other words ifL_{E}Ω ≠ cΩ for any constant $c$.Such a symmetries may be considered as analog of non-Noether symmetriesdefined in Hamiltonian systems and similarly to the Hamiltonian case one can tryto construct conservation laws by means of generator of symmetry $E$and invariant differential form $\Omega $. Namely we have the followingtheorem, which is reformulation of Hojman's theorem in terms of Liouville volume form.

Presence of the non-Noether symmetry not only leads to a sequence of conservation laws, but alsoendows the phase space with a number of interesting geometric structures and it appears that such asymmetry is related to many important concepts used in theory of dynamical systems.One of the such concepts is Lax pair that plays quite important role in constructionof completely integrable models.Let us recall that Lax pair of Hamiltonian system on Poisson manifold $M$ isa pair $(L\; ,\; P)$ of smooth functions on $M$ with values in someLie algebra $g$ such that the time evolution of $L$ is given byadjoint actiond dt L = [L , P] = − ad_{P}L where $[\; ,\; ]$ is a Lie bracket on $g$. It is well known that each Laxpair leads to a number of conservation laws. When $g$ is some matrix Lie algebrathe conservation laws are just traces of powers of $L$I^{(k)} =1 2 Tr(L^{k}) since trace is invariant under coadjoint actiond dt I^{(k)} = 1 2 d dt Tr(L^{k}) =1 2 Tr(d dt L^{k}) = k 2 Tr(L^{k − 1}d dt L) = k 2 Tr(L^{k − 1}[L , P]) = 1 2 Tr([L^{k}, P]) = 0 It is remarkable that each generator of the non-Noethersymmetry canonically leads to the Lax pair of a certain type.Such a Lax pairs have definite geometric origin, their Lax matrices are formedby coefficients of invariant tangent valued 1-form on the phase space.In the local coordinates $z$_{s}, where the bivector field$W$, symplectic form $\omega $ and the generatorof the symmetry $E$ have the following formW = ∑ rs W_{rs}∂ ∂z_{r} ∧ ∂ ∂z_{r} ω = ∑ rs ω_{rs}dz_{r} ∧ dz_{s} E = ∑ s E_{s}∂ ∂z_{s} corresponding Lax pair can be calculated explicitly.Namely we have the following theorem (see also [55]-[56]):

Now let us focus on the integrability issues. We know that$n$ integrals of motion are associated with each generator of non-Noethersymmetry, in the same time we know that, according to the Liouville-Arnold theorem,regular Hamiltonian system $(M,\; h)$ on $2n$ dimensional symplectic manifold$M$ is completely integrable (can be solved completely) if it admits$n$ functionally independent integrals of motion in involution.One can understand functional independence of set of conservation laws$c$_{1}, c_{2} ... c_{n} aslinear independence of either differentials of conservation laws$dc$_{1}, dc_{2} ... dc_{n} orcorresponding Hamiltonian vector fields$X$_{c1}, X_{c2} ... X_{cn}.Strictly speaking we can say that conservation laws $c$_{1}, c_{2} ... c_{n}are functionally independent if Lesbegue measure of the set of points of phase space $M$where differentials $dc$_{1}, dc_{2} ... dc_{n} become linearly dependentis zero. Involutivity of conservation laws means that all possible Poisson brackets ofthese conservation laws vanish pair wise{c_{i} , c_{j}} = 0 i, j = 1... n In terms of the vector fields, existence of involutive family of $n$functionally independent conservation laws$c$_{1}, c_{2} ... c_{n}implies that corresponding Hamiltonian vector fields$X$_{c1}, X_{c2} ... X_{cn}span Lagrangian subspace (isotropic subspace of dimension $n$)of tangent space (at each point of $M$).Indeed, due to property (23){c_{i} , c_{j}} = ω(X_{ci} , X_{cj}) = 0 thus space spanned by $X$_{c1}, X_{c2} ... X_{cn}is isotropic. Dimension of this space is $n$ so it is Lagrangian. Note also that distribution$X$_{c1}, X_{c2} ... X_{cn}is integrable since due to (22)[X_{ci} , X_{cj}] = X_{{ci , cj}} = 0 and according to Frobenius theorem there exists submanifold of $M$ such thatdistribution $X$_{c1}, X_{c2} ... X_{cn} spans tangentspace of this submanifold. Thus for phase space geometry existence of complete involutive setof integrals of motion implies existence of invariant Lagrangian submanifold.

Now let us look at conservation laws $Y(1),\; Y(2)...\; Y(n)$associated with generator of non-Noether symmetry. Generally speaking these conservation laws might appear to be neither functionally independent nor involutive.However it is reasonable to ask the question – what condition should be satisfiedby the generator of the non-Noether symmetry to ensure the involutivity($\{Y(k),\; Y(m)\}\; =\; 0$) of conserved quantities?In Lax theory situation is very similar — each Lax matrix leads to the set ofconservation laws but in general this set is not involutive, however in Lax theorythere is certain condition known as Classical Yang-Baxter Equation (CYBE)that being satisfied by Lax matrix ensures that conservation laws are in involution.Since involutivity of the conservation laws is closely related to the integrability,it is essential to have some analog of CYBE for the generatorof non-Noether symmetry. To address this issue we would like to propose the following theorem.

Further we will focus on non-Noether symmetries that satisfy condition (110). Besidesyielding involutive families of conservation laws, such a symmetries appear to be relatedto many known geometric structures such as bi-Hamiltonian systems [53]and Frölicher-Nijenhuis operators (torsionless tangent valued differential 1-forms).The relationship between non-Noether symmetries and bi-Hamiltonian structures wasalready implicitly outlined in the proof of Theorem 4. Now let us pay more attention tothis issue.

Originally bi-Hamiltonian structures were introduced by F. Magri in analisys ofintegrable infinite dimensional Hamiltonian systems such as Korteweg-de Vries (KdV) andmodified Korteweg-de Vries (mKdV) hierarchies, Nonlinear Schrödinger equationand Harry Dym equation. Since that time bi-Hamiltonian formalism is effectively usedin construction of involutive families of conservation laws in integrable models

Generic bi-Hamiltonian structure on $2n$ dimensional manifold consists outof two Poisson bivector fields $W$ and $\u0174$ satisfying certaincompatibility condition $[\u0174\; ,\; W]\; =\; 0$. If, in addition, one of these bivectorfields is nondegenerate ($Wn\ne \; 0$) then bi-Hamiltonian systemis called regular. Further we will discuss only regular bi-Hamiltonian systems.Note that each Poisson bivector field by definition satisfies condition (15). So we actuallyimpose four restrictions on bivector fields $W$ and $\u0174$[W , W] = [Ŵ , W] = [Ŵ , Ŵ] = 0 andW^{n} ≠ 0 During the proof of Theorem 4 we already showed that bivector fields$W$ and $\u0174\; =\; [E\; ,\; W]$ satisfy conditions (126)(see (112)-(116)), thus we can formulate the following statement

In terms of differential forms bi-Hamiltonian structure is formed by couple ofclosed differential 2-forms: symplectic form $\omega $(such that $d\omega \; =\; 0$ and $\omega n\ne \; 0$)and $\omega \ast =\; L$_{E}ω(clearly $d\omega \ast =\; dL$_{E}ω= L_{E}dω = 0). It is important that by taking Lie derivative ofHamilton's equationi_{Xh}ω + dh = 0 along the generator $E$ of symmetryL_{E}(i_{Xh}ω + dh) =i_{[E , Xh]}ω + i_{Xh}L_{E}ω + L_{E}dh =i_{Xh}ω^{∗} + dL_{E}h =0 one obtains another Hamilton's equationi_{Xh}ω^{∗} + dh^{∗} = 0 where $h\ast =\; L$_{E}h. This is actually second Hamiltonian realizationof equations of motion and thus under certain conditions existence of non-Noether symmetrygives rise to additional presymplectic structure $\omega \ast $and additional Hamiltonian realization of the dynamical system.In many integrable models admitting bi-Hamiltonian realization (including Toda chain,Korteweg-de Vries hierarchy, Nonlinear Schrödinger equation, Broer-Kaup system andBenney system) non-Noether symmetries that are responsible for existence of bi-Hamiltonian structureshas been found and motivated further investigation of relationship betweensymmetries and bi-Hamiltonian structures. Namely it seems to be interesting to knowwhether in general case existence of bi-Hamiltonian structure is related to non-Noether symmetry.Let us consider more general case and suppose that we have couple of differential 2-forms$\omega $ and $\omega \ast $such thatdω = dω^{∗} = 0, ω^{n} ≠ 0 i_{Xh}ω + dh = 0 andi_{Xh}ω^{∗} + dh^{∗} = 0 The question is whether there exists vector field $E$ (generator of non-Noether symmetry)such that $[E\; ,\; X$_{h}] = 0 and$\omega \ast =\; L$_{E}ω.

The answer depends on $\omega \ast $.Namely if $\omega \ast $ is exact form(there exists 1-form $\theta \ast $ such that$\omega \ast =\; d\theta \ast $)then one can argue that such a vector field exists and thus anyexact bi-Hamiltonian structure is related to hidden non-Noethersymmetry. To outline proof of this statement let us introducevector field $E\ast $ defined byi_{E∗}ω = θ^{∗} (such a vector field always exist because $\omega $is nondegenerate 2-form).By constructionL_{E∗} ω = ω^{∗} IndeedL_{E∗}ω = di_{E∗}ω +i_{E∗}dω = dθ^{∗} = ω^{∗} Andi_{[E∗, Xh]}ω =L_{E∗}(i_{Xh}ω)− i_{Xh}L_{E∗}ω =− d(E^{∗}(h)− h^{∗}) = − dh' In other words $[X$_{h} , E^{∗}] is Hamiltonian vector field[X_{h} , E] = X_{h'} One can also construct locally Hamiltonian vector field $X$_{g},that satisfies the same commutation relation. Namely let us definefunction (in general case this can be done only locally)g(z) = t ∫ 0 h'dt where integration along solution of Hamilton's equation, with fixed origin and end point in$z(t)\; =\; z$, is assumed.And then it is easy to verify that locally Hamiltonian vector field associated with $g(z)$,by construction, satisfies the same commutation relations as$E\ast $ (namely $[X$_{h} , X_{g}] = X_{h'}).Using $E\ast $ and $X$_{h'}one can construct generator of non-Noether symmetry —non-Hamiltonian vector field $E\; =\; E\ast -\; X$_{g}commuting with $X$_{h} and satisfyingL_{E}ω = L_{E∗}ω− L_{Xg}ω = L_{E∗}ω = ω^{∗} (thanks to Liouville's theorem $L$_{Xg}ω = 0). So incase of regular Hamiltonian system every exact bi-Hamiltonian structure isnaturally associated with some (non-Noether) symmetry of space of solutions.In case where bi-Hamiltonian structure is not exact($\omega \ast $ is closed but not exact) then due toω^{∗} = L_{E}ω =di_{E}ω + i_{E}dω = di_{E}ω it is clear that such a bi-Hamiltonian system is not related to symmetry.However in all known cases bi-Hamiltonian structures seem to be exact.

Another important concept that is often used in theory of dynamical systems and maybe related to the non-Noether symmetry is the bidifferential calculus (bicomplex approach).Recently A. Dimakis and F. Müller-Hoissenapplied bidifferential calculi to the wide range of integrable modelsincluding KdV hierarchy, KP equation, self-dual Yang-Mills equation,Sine-Gordon equation, Toda models, non-linear Schrödingerand Liouville equations. It turns out that these models can be effectivelydescribed and analyzed using the bidifferential calculi[17], [24].Here we would like to show that each generator of non-Noether symmetrysatisfying condition $[[E[E\; ,\; W]]W]\; =\; 0$ gives rise to certainbidifferential calculus.

Before we proceed let us specify what kind of bidifferential calculi we plan to consider.Under the bidifferential calculus we mean the graded algebra of differential formsover the phase spaceΩ =∞ ∪ k = 0 Ω^{(k)} ($\Omega (k)$ denotes the space of $k$-degree differential forms)equipped with a couple of differential operatorsd, đ : Ω^{(k)} → Ω^{(k + 1)} satisfying conditions$d2=\; \u01112=\; d\u0111\; +\; \u0111d\; =\; 0$ (see [24]). In other words we have two De Rhamcomplexes $M,\; \Omega ,\; d$ and $M,\; \Omega ,\; \u0111$on algebra of differential forms over the phase space. And these complexes satisfycertain compatibility condition — their differentials anticommute with each other$d\u0111\; +\; \u0111d\; =\; 0$.Now let us focus on non-Noether symmetries.It is interesting that if generator of the non-Noether symmetry satisfiesequation $[[E[E\; ,\; W]]W]\; =\; 0$ then we are able to construct an invariantbidifferential calculus of a certain type.This construction is summarized in the following theorem:

Finally we would like to reveal some features of the operator$\u0154$_{E}(89) and to show how Frölicher-Nijenhuis geometry arises inHamiltonian system that possesses certain non-Noether symmetry.From the geometric properties of the tangent valued forms we knowthat the traces of powers of a linear operator $F$on tangent bundle are in involution whenever its Frölicher-Nijenhuis torsion$T(F)$ vanishes, i. e. whenever for arbitrary vector fields $X,Y$ the conditionT(F)(X , Y) = [FX , FY] −F([FX , Y] + [X , FY] − F[X , Y]) = 0 is satisfied.Torsionless forms are also called Frölicher-Nijenhuis operators and are widely used intheory of integrable models, where they play role of recursion operators and are usedin construction of involutive family of conservation laws.We would like to show that each generator of non-Noether symmetry satisfying equation$[[E[E\; ,\; W]]W]\; =\; 0$canonically leads to invariant Frölicher-Nijenhuis operator on tangentbundle over the phase space. This operator can be expressed in terms of generator of symmetryand isomorphism defined by Poisson bivector field. Strictly speaking we have the following theorem.

One-parameter group of transformations $g$_{z}defined by (28) naturally acts on algebra of integrals of motion.Namely for each conservation lawd dt J = 0 one can define one-parameter family of conserved quantities $J(z)$by applying group of transformations $g$_{z} to $J$J(z) = g_{z}(J) = e^{zLE}J =J + zL_{E}J + ½(zL_{E})^{2}J + ... Property (29) ensures that $J(z)$ is conserved for arbitrary valuesof parameter $z$d dt J(z) =d dt g_{z}(J) =g_{z}(d dt J) = 0 and thus each conservation law gives rise to whole family of conservedquantities that form orbit of group of transformations $g$_{a}.

Such an orbit $J(z)$ is called involutive if conservation laws that formit are in involution{J(z_{1}) , J(z_{2})} = 0 (for arbitrary values of parameters $z$_{1}, z_{2}). On $2n$ dimensionalsymplectic manifold each involutive family that contains $n$ functionally independentintegrals of motion naturally gives rise to integrable system (due to Liouville-Arnold theorem).So in order to identify those orbits that may be related to integrable models itis important to know how involutivity of family of conserved quantities $J(z)$is related to properties of initial conserved quantity $J(0)\; =\; J$ and nature ofgenerator $E$ of group $g$_{z} = e^{zLE}.In other words we would like to know what condition must be satisfied by generator ofsymmetry $E$ and integral of motion $J$ to ensure that$\{J(z$_{1}) , J(z_{2})} = 0. To address this issue and to describe class of vector fieldsthat possess nontrivial involutive orbits we would like to propose the followingtheorem

Further we will use this theorem to prove involutivity of family of conservation laws constructed using non-Noether symmetry of Toda chain.

To illustrate features of non-Noether symmetries we oftenrefer to two and three particle non-periodic Toda systems.However it turns out that non-Noether symmetries are present ingeneric n-particle non-periodic Toda chains as well, moreover they preservebasic features of symmetries (53), (61).In case of n-particle Toda model symmetry yields $n$functionally independent conservation laws in involution,gives rise to bi-Hamiltonian structure of Toda hierarchy,reproduces Lax pair of Toda system, endows phase space withFrölicher-Nijenhuis operator and leads to invariantbidifferential calculus on algebra of differential forms over phase spaceof Toda system.

First of all let us remind that Toda model is$2n$ dimensional Hamiltonian system that describes the motionof $n$ particles on the line governed by the exponential interaction.Equations of motion of the non periodic n-particle Toda model ared dt q_{s} = p_{s}d dt p_{s} = ε(s − 1)e^{qs − 1 − qs} −ε(n − s)e^{qs − qs + 1} ($\epsilon (k)\; =\; -\; \epsilon (-\; k)\; =\; 1$ for any natural$k$ and $\epsilon (0)\; =\; 0$) and can be rewritten in Hamiltonian form(24) with canonical Poisson bracket defined by Poisson bivectorW = n ∑ s = 1 ∂ ∂p_{s} ∧ ∂ ∂q_{s} and Hamiltonian equal toh = ½n ∑ s = 1 p_{s}^{2} +n − 1 ∑ s = 1 e^{qs − qs + 1} Note that in two and three particle case we have used slightly different notationsz_{s} = p_{s}z_{n + s} = q_{s} s = 1, 2, (3); n = 2(3) for local coordinates.The group of transformations $g$_{z} generated by the vector field$E$ will be symmetry of Toda chain if for each$p$_{s}, q_{s} satisfying Toda equations(214)$g$_{z}(p_{s}), g_{z}(q_{s})also satisfy it.Substituting infinitesimal transformationsg_{z}(p_{s}) = p_{s} + zE(p_{s}) + O(z^{2})g_{z}(p_{s}) = q_{s} + zE(q_{s}) + O(z^{2}) into (214) and grouping first order terms gives rise to theconditionsd dt E(q_{s}) = E(p_{s})d dt E(p_{s}) = ε(s − 1)e^{qs − 1 − qs}(E(q_{s − 1}) − E(q_{s})) − ε(n − s)e^{qs − qs + 1}(E(q_{s}) − E(q_{s + 1})) One can verify that the vector field defined byE(p_{s}) = ½p_{s}^{2} +ε(s − 1)(n − s + 2)e^{qs − 1 − qs} −ε(n − s)(n − s) e^{qs − qs + 1}+ t 2 (ε(s − 1)(p_{s − 1} + p_{s})e^{qs − 1 − qs} −ε(n − s)(p_{s} + p_{s + 1})e^{qs − qs + 1})E(q_{s}) = (n − s + 1)p_{s} −½s − 1 ∑ k = 1 p_{k}+ ½n ∑ k = s + 1 p_{k}+ t 2 (p_{s}^{2} +ε(s − 1)e^{qs − 1 − qs} +ε(n − s)e^{qs − qs + 1}) satisfies (31) and generates symmetry of Toda chain. It appears that this symmetry is non-Noether since it does notpreserve Poisson bracket structure $[E\; ,\; W]\; \ne \; 0$and additionally one can check that Yang-Baxter equation$[[E[E\; ,\; W]]W]\; =\; 0$ is satisfied.This symmetry may play important role inanalysis of Toda model. First let us note that calculating $L$_{E}Wleads to the following Poisson bivector fieldŴ = [E , W] =n ∑ s = 1 p_{s}∂ ∂p_{s} ∧ ∂ ∂q_{s} + n − 1 ∑ s = 1 e^{qs − qs + 1} ∂ ∂p_{s} ∧ ∂ ∂q_{s + 1} + ∑ r > s ∂ ∂q_{s} ∧ ∂ ∂q_{r} and together $W$ and $L$_{E}W give rise tobi-Hamiltonian structure of Toda model (compare with [30]).Thus bi-Hamiltonian realization of Toda chain can be considered as manifestationof hidden symmetry.In terms of bivector fields these bi-Hamiltonian system is formed byThe conservation laws (45) associated with the symmetry reproduce well knownset of conservation laws of Toda chain.I^{(1)} = C^{(1)} = n ∑ s = 1 p_{s}I^{(2)} = (C^{(1)})^{2} − 2C^{(2)} =n ∑ s = 1 p_{s}^{2} + 2n − 1 ∑ s = 1 e^{qs − qs + 1}I^{(3)} = C^{(1)})^{3} − 3C^{(1)}C^{(2)}+ 3C^{(3)} = n ∑ s = 1 p_{s}^{3} +3n − 1 ∑ s = 1 (p_{s} + p_{s + 1}) e^{qs − qs + 1}I^{(4)} = C^{(1)})^{4} − 4(C^{(1)})^{2}C^{(2)} +2(C^{(2)})^{2} + 4C^{(1)}C^{(3)} − 4C^{(4)}= n ∑ s = 1 p_{s}^{4} + 4n − 1 ∑ s = 1 (p_{s}^{2} + 2p_{s}p_{s + 1} + p_{s + 1}^{2})e^{qs − qs + 1}+ 2n − 1 ∑ s = 1 e^{2(qs − qs + 1)} +4n − 2 ∑ s = 1 e^{qs − qs + 2} I^{(m)} = (− 1)^{m + 1}mC^{(m)} +m − 1 ∑ k = 1 (− 1)^{k + 1}I^{(m − k)}C^{(k)} The condition $[[E[E\; ,\; W]]W]\; =\; 0$ satisfied by generator of thesymmetry $E$ ensures that the conservation laws are in involutioni. e. $\{C(k),\; C(m)\}\; =\; 0$.Thus the conservation laws as well as the bi-Hamiltonian structureof the non periodic Toda chain appear to be associated with non-Noether symmetry.

Using formula (88) one can calculate Lax pairassociated with symmetry (220).Lax matrix calculated in this way has the following non-zero entries(note that in case of $n\; =\; 2$ and $n\; =\; 3$ this formula yields matrices(102)-(105))L_{k, k} = L_{n + k, n + k} = p_{k}L_{n + k, k + 1} = − L_{n + k + 1, k} =ε(n − k)e^{qk − qk + 1}L_{k, n + m} = ε(m − k)m, k = 1, 2, ... , n while non-zero entries of $P$ matrix involved in Lax pair areP_{k, n + k} = 1P_{n + k, k} = − ε(k − 1)e^{qk − 1 − qk}− ε(n − k)e^{qk − qk + 1}P_{n + k, k + 1} = ε(n − k)e^{qk − qk + 1}P_{n + k, k − 1} = ε(k − 1)e^{qk − 1 − qk}k = 1, 2, ... , n This Lax pair constructed from generator of non-Noether symmetryexactly reproduces known Lax pair of Toda chain.

Like two and three particle Toda chain, n-particle Toda model also admitsinvariant bidifferential calculus on algebra of differential forms over the phase space.This bidifferential calculus can be constructed using non-Noether symmetry (see (152)),it consists out of two differential operators $d,\; \u0111$where $d$ is ordinary exterior derivative while $\u0111$can be defined byđq_{s} = p_{s}dq_{s} + ∑ r > s dp_{r} − ∑ s > r dp_{r}đp_{s} = p_{s}dp_{s} − e^{qs − qs + 1}dq_{s + 1}+ e^{qs − 1 − qs}dq_{s} and is extended to whole De Rham complex by linearity, derivation property andcompatibility property $d\u0111\; +\; \u0111d\; =\; 0$.By direct calculations one can verify that calculus constructed in this wayis consistent and satisfies $\u01112=\; 0$ property.One can also check that conservation laws (222) form Lenard scheme(k + 1)đI^{(k)} = kdI^{(k + 1)}

Further let us focus on Frölicher-Nijenhuis geometry. Using formula (173)one can construct invariant Frölicher-Nijenhuis operator, out of generator of non-Noethersymmetry of Toda chain. Operator constructed in this way has the formŔ_{E} = n ∑ s = 1 p_{s}(dp_{s} ⊗ ∂ ∂q_{s} + dq_{s} ⊗ ∂ ∂p_{s} ) − n − 1 ∑ s = 1 e^{qs − qs + 1}dq_{s + 1} ⊗ ∂ ∂p_{s} + n − 1 ∑ s = 1 e^{qs − 1 − qs}dq_{s} ⊗ ∂ ∂p_{s} − ∑ s > r (dp_{s} ⊗ ∂ ∂q_{r} − dp_{r} ⊗ ∂ ∂q_{s} ) One can check that Frölicher-Nijenhuis torsion of this operator vanishes andit plays role of recursion operator for n-particle Toda chain in sense that conservation laws$I(k)$ satisfy recursion relation(k + 1)R_{E}(dI^{(k)}) = kdI^{(k + 1)} Thus non-Noether symmetry of Toda chain not only leads ton functionally independent conservation laws in involution, but alsoessentially enriches phase space geometry by endowing it withinvariant Frölicher-Nijenhuis operator, bi-Hamiltonian system,bicomplex structure and Lax pair.

Finally, in order to outline possible applications of Theorem 8 let us studyaction of non-Noether symmetry (220) on conserved quantitiesof Toda chain. Vector field $E$ defined by (220) generatesone-parameter group of transformations (28) that maps arbitraryconserved quantity $J$ toJ(z) = J + zJ^{(1)} + z^{2} 2! J^{(2)} +z^{3} 3! J^{(3)} + ⋯ whereJ^{(m)} = (L_{E})^{m}J In particular let us focus on family of conserved quantities obtained by action of$g$_{a} = e^{aLE} on total momenta of Toda chainJ = n ∑ s = 1 p_{s} By direct calculations one can check that family $J(z)$, that forms orbitof non-Noether symmetry generated by $E$, reproduces entire involutivefamily of integrals of motion (222). NamelyJ^{(1)} = L_{E}J = ½ n ∑ s = 1 p_{s}^{2} + n − 1 ∑ s = 1 e^{qs − qs + 1}J^{(2)} = L_{E}J^{(1)} = (L_{E})^{2}J =1 2 n ∑ s = 1 p_{s}^{3} +3 2 n − 1 ∑ s = 1 (p_{s} + p_{s + 1})e^{qs − qs + 1}J^{(3)} = L_{E}J^{(2)} = (L_{E})^{3}J =¾ n ∑ s = 1 p_{s}^{4} +3n − 1 ∑ s = 1 (p_{s}^{2} + 2p_{s}p_{s + 1} +p_{s + 1}^{2})e^{qs − qs + 1}+ 3 2 n − 1 ∑ s = 1 e^{2(qs − qs + 1)} +3n − 2 ∑ s = 1 e^{qs − qs + 2}J^{(m)} = L_{E}J^{(m − 1)} = (L_{E})^{m}J

Involutivity of this set of conservation laws can be verified using Theorem 8.In particular one can notice that differential 1-form $s$ defined byE = W(s) (where $E$ is generator of non-Noether symmetry (220))satisfies condition[W[W(s),W](s)] = 3[W(s)[W(s) ,W]] while conservation law $J$ defined by (231)has propertyW(L_{W(s)}dJ) = − [W(s),W](dJ) and thus according to Theorem 8 conservation laws (232)are in involution.

Toda model provided good example of finite dimensional integrable Hamiltonian systemthat possesses non-Noether symmetry. However there are manyinfinite dimensional integrable Hamiltonian systems and in this case inorder to ensure integrability one should constructinfinite number of conservation laws. Fortunately in several integrable modelsthis task can be effectively simplified by identifying appropriate non-Noether symmetry.First let us consider well known infinite dimensional integrable Hamiltonian system –Korteweg-de Vries equation (KdV). The KdV equation has the following formu_{t} + u_{xxx} + uu_{x} = 0 (here $u$ is smooth function of $(t,\; x)\; \in \; R2$).The generators of symmetries of KdV should satisfy conditionE(u)_{t} + E(u)_{xxx} +u_{x}E(u) + uE(u)_{x} = 0 which is obtained by substituting infinitesimal transformation$u\; \to \; u\; +\; zE(u)\; +\; O(z2)$ into KdV equation and grouping first order terms.

Later we will focus on the symmetry generated by the following vector fieldE(u) = 2u_{xx} + 2 3 u^{2} + 1 6 u_{x}v +x 2 (u_{xxx} + uu_{x}) − t 4 (6u_{xxxxx} + 20u_{x}u_{xx} +10 uu_{xxx} + 5u^{2}u_{x}) (here $v$ is defined by $v$_{x} = u).

If $u$ is subjected to zero boundary conditions $u(t,\; -\; \infty )\; =\; u(t,\; +\; \infty )\; =\; 0$then KdV equation can be rewritten in Hamiltonian form u_{t} = {h , u} with Poisson bivector field equal toW = + ∞ ∫ − ∞ dx δ δu ∧ {δ δu }_{x} and Hamiltonian defined byh = + ∞ ∫ − ∞ (u_{x}^{2} − u^{3} 3 ) dx By taking Lie derivative of thesymplectic form along the generator of the symmetry one getssecond Poisson bivector [E , W] = W = + ∞ ∫ − ∞ dx ({δ δu }_{xx} ∧ {δ δu }_{x}+ 2 3 uδ δu ∧ {δ δu }_{x}) involved in bi-Hamiltonian structure of KdV hierarchy andproposed by Magri [58].

Now let us show how non-Noether symmetry can be used to construct conservation lawsof KdV hierarchy. By integrating KdV it is easy to show thatJ^{(0)} = + ∞ ∫ − ∞ u dx is conserved quantity. At the same time Lie derivative of any conservedquantity along generator of symmetry is conserved as well,while taking Lie derivative of $J(0)$ along $E$ gives rise toinfinite sequence of conservation laws $J(m)=\; (L$_{E})^{m}J^{(0)}that reproduce well known conservation laws of KdV equationJ^{(0)} = + ∞ ∫ − ∞ u dxJ^{(1)} = L_{E}J^{(0)} =¼+ ∞ ∫ − ∞ u^{2} dx J^{(2)} = (L_{E})^{2}J^{(0)} =5 8 + ∞ ∫ − ∞ (u^{3} 3 − u_{x}^{2}) dx J^{(3)} = (L_{E})^{3}J^{(0)} = 35 16 + ∞ ∫ − ∞ (5 36 u^{4} −5 3 uu_{x}^{2} + u_{xx}^{2}) dxJ^{(m)} = (L_{E})^{m}J^{(0)} Thus the conservation laws and bi-Hamiltonian structures of KdV hierarchy are related to the non-Noether symmetry of KdV equation.

Among nonlinear partial differential equations that describe propagation of waves in shallow waterthere are many remarkable integrable systems. We have already discussed case of KdV equation, that possess non-Noether symmetries leading to the infinite sequence of conservation lawsand bi-Hamiltonian realization of these equations,now let us consider other important water wave systems.It is reasonable to start with dispersive water wave system [73],[74],since many other models can be obtained from it by reduction.Evolution of dispersive water wave system is governed bythe following set of equationsu_{t} = u_{x}w + uw_{x}v_{t} = uu_{x} − v_{xx} + 2v_{x}w + 2vw_{x}w_{t} = w_{xx} − 2v_{x} + 2ww_{x} Each symmetry of this system must satisfy linear equationE(u)_{t} = (wE(u))_{x} + (uE(w))_{x}E(v)_{t} = (uE(u))_{x} − E(v)_{xx} + 2(wE(v))_{x} + 2(vE(w))_{x}E(w)_{t} = E(w)_{xx} − 2E(v)_{x} + 2(wE(w))_{x} obtained by substituting infinitesimal transformationsu → u + zE(u) + O(z^{2})v → v + zE(v) + O(z^{2})w → w + zE(w) + O(z^{2}) into equations of motion (245) and grouping first order(in $a$) terms. One of the solutions of this equation yieldsthe following symmetry of dispersive water wave systemE(u) = uw + x(uw)_{x} + 2t(uw^{2} − 2uv + uw_{x})_{x}E(v) = 3 2 u^{2} + 4vw − 3v_{x} + x(uu_{x} + 2(vw)_{x} − v_{xx})+ 2t(u^{2}w − uu_{x} − 3v^{2} + 3vw^{2} − 3v_{x}w + v_{xx})_{x}E(w) = w^{2} + 2w_{x} − 4v + x(2ww_{x} + w_{xx} − 2v_{x})− 2t(u^{2} + 6vw − w^{3} − 3ww_{x} − w_{xx})_{x} and it is remarkable that this symmetry is local in sense that $E(u)$ in point$x$ depends only on $u$ and its derivatives evaluated in the same point,(this is not the case in KdV where symmetry is non localdue to presence of non local field $v$ defined by $v$_{x} = u).

Before we proceed let us note that dispersive water wave system is actually infinite dimensionalHamiltonian dynamical system. Assuming that $u,\; v$ and $w$ fieldsare subjected to zero boundary conditionsu(± ∞) = v(± ∞) = w(± ∞) = 0 it is easy to verify that equations (245) can be represented in Hamiltonian formu_{t} = {h , u}v_{t} = {h , v}w_{t} = {h , w} with Hamiltonian equal toh = − ¼ + ∞ ∫ − ∞ (u^{2}w + 2vw^{2} − 2v_{x}w − 2v^{2})dx and Poisson bracket defined by the following Poisson bivector fieldW = + ∞ ∫ − ∞ {½ δ δu ∧ {δ δu }_{x} +δ δv ∧ {δ δw }_{x}} dx Now using our symmetry that appears to be non-Noether, one can calculate second Poissonbivector field involved in the bi-Hamiltonian realization of dispersive water wave systemŴ = [E , W] = − 2 + ∞ ∫ − ∞ {u δ δv ∧ {δ δu }_{x}+ v δ δv ∧ {δ δv }_{x}+ {δ δv }_{x} ∧ {δ δw }_{x}+ w δ δv ∧ {δ δw }_{x}+ {δ δw }_{x} ∧ δ δw } dx Note that $\u0174$ give rise to the second Hamiltonian realization ofthe modelu_{t} = {h^{∗} , u}_{∗}v_{t} = {h^{∗} , v}_{∗}w_{t} = {h^{∗} , w}_{∗} whereh^{∗} = − ¼ + ∞ ∫ − ∞ (u^{2} + 2vw)dx and $\{\; ,\; \}$_{∗} is Poisson bracket defined bybivector field $\u0174$.

Now let us pay attention to conservation laws. By integrating third equationof dispersive water wave system (245) it is easy to show thatJ^{(0)} =+ ∞ ∫ − ∞ wdx is conservation law. Using non-Noether symmetryone can construct other conservation laws by taking Lie derivativeof $J(0)$ along the generator of symmetry and in this wayentire infinite sequence of conservation laws of dispersive water wave systemcan be reproducedJ^{(0)} = + ∞ ∫ − ∞ wdxJ^{(1)} = L_{E}J^{(0)} = − 2 + ∞ ∫ − ∞ vdx J^{(2)} = L_{E}J^{(1)} = (L_{E})^{2}J^{(0)} =− 2+ ∞ ∫ − ∞ (u^{2} + 2vw)dxJ^{(3)} = L_{E}J^{(2)} = (L_{E})^{3}J^{(0)} =− 6+ ∞ ∫ − ∞ (u^{2}w + 2vw^{2} − 2v_{x}w − 2v^{2})dxJ^{(4)} = L_{E}J^{(3)} = (L_{E})^{4}J^{(0)}= − 24 + ∞ ∫ − ∞ (u^{2}w^{2} + u^{2}w_{x} − 2u^{2}v − 6v^{2}w +2vw^{3} − 3v_{x}w^{2} − 2v_{x}w_{x})dxJ^{(n)} = L_{E}J^{(n − 1)} = (L_{E})^{n}J^{(0)} Thus conservation laws and bi-Hamiltonian structure of dispersive waterwave system can be constructed by means of non-Noether symmetry.

Note that symmetry (248) can be used in many otherpartial differential equations that can be obtained by reduction from dispersivewater wave system. In particular one can use it in dispersiveless water wave system,Broer-Kaup system, dispersiveless long wave system, Burger's equation etc.In case of dispersiveless water waves systemu_{t} = u_{x}w + uw_{x}v_{t} = uu_{x} + 2v_{x}w + 2vw_{x}w_{t} = − 2v_{x} + 2ww_{x} symmetry (248) is reduced toE(u) = uw + x(uw)_{x} + 2t(uw^{2} − 2uv)_{x}E(v) = 3 2 u^{2} + 4vw + x(uu_{x} + 2(vw)_{x})+ 2t(u^{2}w − 3v^{2} + 3vw^{2})_{x}E(w) = w^{2} − 4v + x(2ww_{x} − 2v_{x}) − 2t(u^{2} + 6vw − w^{3})_{x} and corresponding conservation laws (257) reduce toJ^{(0)} = + ∞ ∫ − ∞ wdxJ^{(1)} = L_{E}J^{(0)} =− 2 + ∞ ∫ − ∞ vdxJ^{(2)} = L_{E}J^{(1)} = (L_{E})^{2}J^{(0)} =− 2 + ∞ ∫ − ∞ (u^{2} + 2vw)dxJ^{(3)} = L_{E}J^{(2)} = (L_{E})^{3}J^{(0)} =− 6 + ∞ ∫ − ∞ (u^{2}w + 2vw^{2} − 2v^{2})dxJ^{(4)} = L_{E}J^{(3)} = (L_{E})^{4}J^{(0)} =− 24 + ∞ ∫ − ∞ (u^{2}w^{2} − 2u^{2}v − 6v^{2}w + 2vw^{3})dxJ^{(n)} = L_{E}J^{(n − 1)} = (L_{E})^{n}J^{(0)}

Another important integrable model that can be obtained from dispersive water wave systemis Broer-Kaup system [73],[74]v_{t} = ½ v_{xx} + v_{x}w + vw_{x}w_{t} = − ½ w_{xx} + v_{x} + ww_{x} One can check that symmetry (248) of dispersive water wave system,after reduction, reproduces non-Noether symmetry of Broer-Kaup modelE(v) = 4vw + 3v_{x} + x(2(vw)_{x} + v_{xx})+ t(3v^{2} + 3vw^{2} + 3v_{x}w + v_{xx})_{x}E(w) = w^{2} − 2w_{x} + 4v + x(2ww_{x} − w_{xx} + 2v_{x})+ t(6vw + w^{3} − 3ww_{x} + w_{xx})_{x} and gives rise to the infinite sequence of conservation laws of Broer-Kaup hierarchyJ^{(0)} = + ∞ ∫ − ∞ wdxJ^{(1)} = L_{E}J^{(0)} = 2 + ∞ ∫ − ∞ vdxJ^{(2)} = L_{E}J^{(1)} = (L_{E})^{2}J^{(0)} =4 + ∞ ∫ − ∞ vwdxJ^{(3)} = L_{E}J^{(2)} = (L_{E})^{3}J^{(0)} =12 + ∞ ∫ − ∞ (vw^{2} + v_{x}w + v^{2})dxJ^{(4)} = L_{E}J^{(3)} = (L_{E})^{4}J^{(0)} =24 + ∞ ∫ − ∞ (6v^{2}w + 2vw^{3} + 3v_{x}w^{2} − 2v_{x}w_{x})dxJ^{(n)} = L_{E}J^{(n − 1)} = (L_{E})^{n}J^{(0)}

And exactly like in the dispersive water wave system one can rewrite equations of motion(261) in Hamiltonian formv_{t} = {h , v}w_{t} = {h , w} where Hamiltonian ish = ½ + ∞ ∫ − ∞ (vw^{2} + v_{x}w + v^{2})dx while Poisson bracket is defined by the Poisson bivector fieldW = + ∞ ∫ − ∞ {δ δv ∧ {δ δw }_{x}} dx And again, using symmetry (262) one can recover second Poissonbivector field involved in the bi-Hamiltonian realization of Broer-Kaup systemby taking Lie derivative of (266)Ŵ = [E , W] = − 2 + ∞ ∫ − ∞ {v δ δv ∧ {δ δv }_{x}− {δ δv }_{x} ∧ {δ δw }_{x} + w δ δv ∧ {δ δw }_{x}+ δ δw ∧ {δ δw }_{x}} dx This bivector field give rise to the second Hamiltonian realization ofthe Broer-Kaup systemv_{t} = {h^{∗} , v}_{∗}w_{t} = {h^{∗} , w}_{∗} withh^{∗} = −¼ + ∞ ∫ − ∞ vwdx So the non-Noether symmetry of Broer-Kaup system yields infinite sequenceof conservation laws of Broer-Kaup hierarchy and endows it with bi-Hamiltonian structure.

By suppressing dispersive terms in Broer-Kaup system one reduces it to more simpleintegarble model — dispersiveless long wave system [73],[74]v_{t} = v_{x}w + vw_{x}w_{t} = v_{x} + ww_{x} in this case symmetry (248) reduces to more simple non-Noether symmetryE(v) = 4vw + 2x(vw)_{x} + 3t(v^{2} + vw^{2})_{x}E(w) = w^{2} + 4v + 2x(ww_{x} + v_{x}) + t(6vw + w^{3})_{x} while the conservation laws of Broer-Kaup hierarchy reduce tosequence of conservation laws of dispersiveless long wave systemJ^{(0)} = + ∞ ∫ − ∞ wdxJ^{(1)} = L_{E}J^{(0)} = 2 + ∞ ∫ − ∞ vdxJ^{(2)} = L_{E}J^{(1)} = (L_{E})^{2}J^{(0)} =4 + ∞ ∫ − ∞ vwdxJ^{(3)} = L_{E}J^{(2)} = (L_{E})^{3}J^{(0)} =12 + ∞ ∫ − ∞ (vw^{2} + v^{2})dxJ^{(4)} = L_{E}J^{(3)} = (L_{E})^{4}J^{(0)} =48 + ∞ ∫ − ∞ (3v^{2}w + vw^{3})dxJ^{(n)} = L_{E}J^{(n − 1)} = (L_{E})^{n}J^{(0)}

At the same time bi-Hamitonian structure of Broer-Kaup hierarchy, after reductiongives rise to bi-Hamiltonian structure of dispersiveless long wave systemW = + ∞ ∫ − ∞ {δ δv ∧ {δ δw }_{x}} dxŴ = [E , W] = − 2 + ∞ ∫ − ∞ {v δ δv ∧ {δ δv }_{x}+ w δ δv ∧ {δ δw }_{x} + δ δw ∧ {δ δw }_{x}} dx

Among other reductions of dispersive water wave system one should probably mentionBurger's equation [73],[74]w_{t} = w_{xx} + ww_{x} However Hamiltonian realization of this equation is unknown(for instance Poisson bivector field of dispersive water wave system(252) vanishes during reduction).

Now let us consider another integrable system of nonlinear partialdifferential equations — Benney system [73],[74]. Time evolution of this dynamicalsystem is governed by equations of motion u_{t} = vv_{x} + 2(uw)_{x}v_{t} = 2u_{x} + (vw)_{x}w_{t} = 2v_{x} + 2ww_{x} To determine symmetries of the system one has to look for solutions oflinear equationE(u)_{t} = (vE(v))_{x} + 2(uE(w))_{x} + 2(wE(u))_{x}E(v)_{t} = 2E(u)_{x} + (vE(w))_{x} + (wE(v))_{x}E(w)_{t} = 2E(v)_{x} + 2(wE(w))_{x} obtained by substituting infinitesimal transformationsu → u + zE(u) + O(z^{2})v → v + zE(v) + O(z^{2})w → w + zE(w) + O(z^{2}) into equations (275) and grouping first order terms.In particular one can check that the vector field $E$ defined byE(u) = 5uw + 2v^{2} + x(2(uw)_{x} + vv_{x}) + 2t(4uv + v^{2}w + 3uw^{2})_{x}E(v) = vw + 6u + x((vw)_{x} + 2u_{x}) + 2t(4uw + 3v^{2} + vw^{2})_{x}E(w) = w^{2} + 4v + 2x(ww_{x} + v_{x}) + 2t(w^{3} + 4vw + 4u)_{x} satisfies equation (276) and therefore generates symmetry of Benney system.The fact that this symmetry is local simplifies further calculations.

At the same time, it is known fact, that under zero boundary conditionsu(± ∞) = v(± ∞) = w(± ∞) = 0 Benney equations can be rewritten in Hamiltonian formu_{t} = {h , u}v_{t} = {h , v}w_{t} = {h , w} with Hamiltonianh = − ½ + ∞ ∫ − ∞ (2uw^{2} + 4uv + v^{2}w)dx and Poisson bracket defined by the following Poisson bivector fieldW = + ∞ ∫ − ∞ {½ δ δv ∧ {δ δv }_{x}+ δ δu ∧ {δ δw }_{x}} dx Using symmetry (278) that in fact is non-Noether one, we can reproducesecond Poisson bivector field involved in the bi-Hamiltonian structure of Benney hierarchy(by taking Lie derivative of $W$ along $E$)Ŵ = [E , W] = − 3 + ∞ ∫ − ∞ u δ δu ∧ {δ δu }_{x}+ v δ δu ∧ {δ δv }_{x}+ w δ δu ∧ {δ δw }_{x}+ 2 δ δv ∧ {δ δw }_{x}} dx Poisson bracket defined by bivector field $\u0174$ gives riseto the second Hamiltonian realization of Benney systemu_{t} = {h^{∗} , u}_{∗}v_{t} = {h^{∗} , v}_{∗}w_{t} = {h^{∗} , w}_{∗} with new Hamiltonianh^{∗} = 1 6 + ∞ ∫ − ∞ (v^{2} + 2uw)dx Thus symmetry (278) is closely related tobi-Hamiltonian realization of Benney hierarchy.

The same symmetry yields infinite sequence of conservation laws of Benney system.Namely one can construct sequence of integrals of motion by applying non-Noethersymmetry (278) toJ^{(0)} = + ∞ ∫ − ∞ wdx (the fact that $J(0)$ is conserved can be verified by integratingthird equation of Benney system). The sequence looks likeJ^{(0)} = + ∞ ∫ − ∞ wdxJ^{(1)} = L_{E}J^{(0)} =2 + ∞ ∫ − ∞ vdxJ^{(2)} = L_{E}J^{(1)} = (L_{E})^{2}J^{(0)} =8 + ∞ ∫ − ∞ udxJ^{(3)} = L_{E}J^{(2)} = (L_{E})^{3}J^{(0)} =12 + ∞ ∫ − ∞ (v^{2} + 2uw)dxJ^{(4)} = L_{E}J^{(3)} = (L_{E})^{4}J^{(0)} =48 + ∞ ∫ − ∞ (2uw^{2} + 4uv + v^{2}w)dxJ^{(5)} = L_{E}J^{(4)} = (L_{E})^{5}J^{(0)} =240 + ∞ ∫ − ∞ (4u^{2} + 8uvw + 2uw^{3} + 2v^{3} + v^{2}w^{2})dxJ^{(n)} = L_{E}J^{(n − 1)} = (L_{E})^{n}J^{(0)} So conservation laws and bi-Hamiltonian structure of Benney hierarchyare closely related to its symmetry, that can play important role in analysis ofBenney system and other models that can be obtained from it by reduction.

The fact that many important integrable models, such as Korteweg-de Vries equation, Broer-Kaup system, Benney system and Toda chain,possess non-Noether symmetries that can be effectively usedin analysis of these models, inclines us to think that non-Noether symmetries can playessential role in theory of integrable systems and properties of this class of symmetriesshould be investigated further.The present review indicates that in many cases non-Noether symmetries lead to maximal involutivefamilies of functionally independent conserved quantities and in this way ensure integrabilityof dynamical system. To determine involutivity of conservation laws in cases when it can not be checkedby direct computations (for instance one can not check directly the involutivityin many generic n-dimensional models like Toda chainand infinite dimensional models like KdV hierarchy)we propose analog of Yang-Baxter equation, that being satisfied bygenerator of symmetry, ensures involutivity of familyof conserved quantities associated with this symmetry.

Another important feature of non-Noether symmetries is their relationship withseveral essential geometric concepts, emerging in theory of integrable systems, such asFrölicher-Nijenhuis operators, Lax pairs, bi-Hamiltonian structures andbicomplexes. On the one hand this relationship enlarges possible scope ofapplications of non-Noether symmetries in Hamiltonian dynamics and on the other hand itindicates that existence of invariant Frölicher-Nijenhuis operators,bi-Hamiltonian structures and bicomplexes in many cases can be considered as manifestationof hidden symmetries of dynamical system.

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