You are an expert at summarizing long articles. Proceed to summarize the following text: 
the camassa holm equation was introduced in  @xcite in the form @xmath1 and was intensively explored afterwards ( see , for example , refs . its superizations were also constructed , see  @xcite . since   is not an evolution equation , its integrability properties ( existence and even definition of hamiltonian structures , conservation laws , etc . ) are not standard to establish . one of the ways widely used to overcome this difficulty is to introduce a new unknown  @xmath2 and transform eq .   to the system @xmath3 which has _ almost _ evolutionary form . we stress this `` almost '' , because the second equation in   ( that can be considered as a constrain to the first one ) disrupts the picture and , at best , necessitates to invert the operator  @xmath4 . at worst , dealing with eq .   as with an evolution equation may lead to fallacious results .    in our approach based on the geometrical framework exposed in ref . @xcite , we treat the equation at hand as a submanifold in the manifold of infinite jets and consider two natural extensions of this equation , cf .  with ref  @xcite . the first one is called the _ @xmath5_-covering and serves the role of the tangent bundle . the second extension , @xmath6-covering , is the counterpart to the cotangent bundle . the key property of these extensions is that the spaces of their nonlocal ( in the sense of  @xcite ) symmetries and cosymmetries contain all essential integrability invariants of the initial equation . the efficiency of the method was tested for a number of problems ( see refs . @xcite ) and we apply it to the camassa  holm equation here .    in section  [ sec : underlying - theory ] we briefly expose the necessary definition and facts . section  [ sec : matrix - version ] contains computations for the camassa holm equation in its matrix version ( computations and results are more compact in this representation ) , while in section  [ sec : scalar - version ] we reformulate them for the original form   and compare later the results obtained for the two alternative presentations . finally , section  [ sec : conclusion ] contains discussion of the results obtained . throughout our exposition we use a very stimulating conceptual parallel between categories of smooth manifolds and differential equations proposed initially by a.m.  vinogradov and in its modern form presented in table  [ tab:1 ] . l|l * manifolds * & * * equation**s + smooth manifold & infinitely prolonged equation + point & formal solution + smooth function & conservation law + vector field & higher symmetry + differential @xmath7-form & cosymmetry + de  rham complex & @xmath8st line of vinogradov s @xmath9-spectral sequence + tangent bundle & @xmath5-covering + cotangent bundle & @xmath6-covering +    this table is not just a toy dictionary but a quite helpful tool to formulate important definitions and results . for example , a bivector on a smooth manifold  @xmath10 may be understood as a derivation of the ring  @xmath11 with values in  @xmath12 . translating this statement to the language of differential equations we come to the definition of variational bivectors and their description as shadows of symmetries in the @xmath6-covering ( see theorem  [ thm : ellstar - shad ] below ) . another example : any vector field ( differential @xmath7-form ) on  @xmath10 may be treated as a function on  @xmath13 ( on  @xmath14 ) . hence , to any symmetry ( cosymmetry ) there corresponds a conservation law on the space of the @xmath6-covering ( @xmath5-covering ) . this leads to the notions of nonlocal vectors and forms that , in turn , provide a basis to construct weakly nonlocal structures ( see subsections  [ sec : nonlocal - forms-1 ] and  [ sec : nonlocal - vectors-1 ] ) . of course , these parallels are not completely straightforward ( in technical aspects , especially ) , but extremely enlightening and fruitful . the idea of this paper arose in the discussions one of the authors had with volodya roubtsov in 2007 . we agreed to write two parallel texts on integrability of the camassa holm equation that reflect our viewpoints . the reader can now compare our results with the ones presented in  @xcite . we present here a concise exposition of the theoretical background used in the subsequent sections , see refs .  @xcite . let  @xmath15 be a fiber bundle and  @xmath16 be the bundle of its infinite jets . to simplify our exposition we shall assume that  @xmath17 is a vector bundle . in all applications below  @xmath17 is the trivial bundle  @xmath18 . we consider _ infinite prolongations _ of differential equations as submanifolds  @xmath19 and retain the notation  @xmath20 for the restriction  @xmath21 . any such a manifold is endowed with the _ cartan distribution _ which spans at every point tangent spaces to the graphs of jets . a _ symmetry _ of  @xmath22 is a vector field that preserves this distribution . the set of symmetries is a lie algebra over  @xmath23 denoted by  @xmath24 .    for any equation  @xmath22 its _ linearization operator _ @xmath25 is defined , where  @xmath26 is the module of smooth functions on  @xmath22 . ] of sections of the pullback  @xmath27 and  @xmath28 is the module of sections of some vector bundle over  @xmath22 . then  @xmath24 can be identified with solutions of the equation @xmath29 for two symmetries  @xmath30 , @xmath31 their commutator is denoted by  @xmath32 . denote by  @xmath33 the module of horizontal @xmath34-forms on  @xmath22 and introduce the notation @xmath35 for any module  @xmath36 . adjoint _ to  @xmath37 operator @xmath38 arises and solutions of the equation @xmath39 are called _ cosymmetries _ of  @xmath22 ; the space of cosymmetries is denoted by  @xmath40 . let  @xmath41 be the _ horizontal de  rham differential_. a _ conservation law _ of the equation  @xmath22 is a closed form  @xmath42 . to any conservation law there corresponds its _ generating function _ @xmath43 , where  @xmath44 is the differential in the @xmath45 term of vinogradov s @xmath9-spectral sequence , see  @xcite . in the evolutionary case  @xmath46 coincides with the euler  lagrange operator . a conservation law is _ trivial _ if its generating function vanishes . in particular , @xmath47-exact conservation laws are trivial .    a vector field on  @xmath22 is called a _ @xmath9-field _ if it lies in the cartan distribution . a differential operator  @xmath48 , @xmath28 and  @xmath36 being @xmath49-modules , is called a _ @xmath9-differential operator _ if it is locally expressed in terms of @xmath9-fields . for example ,  @xmath37 is a @xmath9-differential operator . a @xmath9-differential operator  @xmath50 is said to be a _ variational bivector _ on  @xmath22 if @xmath51 this condition means that  @xmath52 takes cosymmetries of  @xmath22 to symmetries . if  @xmath22 is an evolution eq .   implies also that  @xmath53 . a bivector  @xmath52 is a _ _ hamiltonian structure _ on  @xmath22 if  @xmath54\!]}^{\mathrm{s}}=0 $ ] , where  @xmath55\!]}^{\mathrm{s}}$ ] is the _ variational schouten bracket _ ( see also  @xcite ) . two hamiltonian structures are _ compatible _ if  @xmath56\!]}^{\mathrm{s}}=0 $ ] . a @xmath9-differential operator  @xmath57 is called a _ variational @xmath58-form on  @xmath22 _ if @xmath59 such operators take symmetries to cosymmetries and in evolutionary case are skew - adjoint . they are elements of the term  @xmath60 of vinogradov s @xmath9-spectral sequence . a variational form is a _ symplectic structure _ on the equation  @xmath22 if it is _ variationally closed _ , i.e. ,  @xmath61 , where  @xmath62 is the corresponding differential . we shall also consider _ recursion @xmath9-differential operators _  @xmath63 and  @xmath64 satisfying the conditions @xmath65 for some @xmath9-differential operators  @xmath66 and  @xmath67 . an operator  @xmath68 satisfies the _ nijenhuis condition _ if  @xmath69\!]}^{\mathrm{n}}=0 $ ] , where  @xmath55\!]}^{\mathrm{n}}$ ] is the _ variational nijenhuis bracket_. a recursion operator  @xmath68 is _ compatible _ with a hamiltonian structure  @xmath52 is the composition  @xmath70 is a hamiltonian structure as well . let  @xmath22 and  @xmath71 be equations and  @xmath72 be a fiber bundle . denote by  @xmath9 and  @xmath73 the cartan distributions on  @xmath22 and  @xmath71 , resp . we say that  @xmath74 is a _ covering _ if for any  @xmath75 the differential  @xmath76 isomorphically maps  @xmath77 onto  @xmath78 . a particular case of coverings ( the so - called _ abelian _ coverings ) is naturally associated with closed horizontal @xmath7-formswhen  @xmath79 , abelian coverings are associated with conservation laws of the equation  @xmath22 . ] .    by definition , any @xmath9-field  @xmath80 on  @xmath22 can be uniquely lifted to a @xmath9-field  @xmath81 on  @xmath71 such that  @xmath82 . consequently , any @xmath9-differential operator  @xmath83 is extended to a @xmath9-differential operator @xmath84 @xmath85 being the algebra of smooth functions on  @xmath71 . a _ nonlocal _ @xmath74-(co)symmetry of  @xmath22 is a ( co)symmetry of the covering equation  @xmath71 . they are solutions of the equations  @xmath86 and  @xmath87 , resp . along with these two equations one can consider the equations @xmath88 their solutions are called _ @xmath74-shadows _ of symmetries and cosymmetries , resp . a shadow of symmetry is a derivation  @xmath89 that preserves the cartan distributions . for any two shadows of symmetries  @xmath30 and  @xmath90 their commutator  @xmath32 can be defined . this commutator is a shadow in a new covering that is canonically determined by  @xmath30 and  @xmath90 . let  @xmath19 be an equation . its _ @xmath5-covering _ @xmath91 is obtained by adding to  @xmath22 the equation  @xmath92 , where  @xmath93 is a new variable . dually , the _ @xmath6-covering _  @xmath94 is constructed by adding the equation  @xmath95 with a new variable  @xmath96 . they are the exact counterparts of the tangent and cotangent bundles in the category of differential equations . by the reasons that will become clear later , we regard both  @xmath93 and  @xmath96 as _ odd _ variables . the main point of our method is the fundamental relation between integrability invariants of  @xmath22 and shadows in  @xmath97 and  @xmath98 . to formulate this relation , let us give an auxiliary definition : for an arbitrary operator equation  @xmath99 we say that  @xmath100 is a _ trivial solutions _ if  @xmath100 is of the form  @xmath101 . classes of solutions modulo trivial ones will be called _ nontrivial solutions_. then the following results hold . [ thm : ell - shad ] there is a one - to - one correspondence between nontrivial solutions of the equation @xmath102 and @xmath97-shadows of symmetries linear w.r.t . the variables  @xmath93 . in a similar way there is a one - to - one correspondence between nontrivial solutions of the equation @xmath103 and @xmath97-shadows of cosymmetries linear w.r.t . the variables  @xmath93 . [ thm : ellstar - shad ] there is a one - to - one correspondence between nontrivial solutions of the equation @xmath104 and @xmath98-shadows of symmetries linear w.r.t . the variables  @xmath96 . in a similar way there is a one - to - one correspondence between nontrivial solutions of the equation @xmath105 and @xmath98-shadows of cosymmetries linear w.r.t . the variables  @xmath96 . [ thm : brackets ] let  @xmath106 and  @xmath107 be recursion operators for symmetries on  @xmath22 and  @xmath108 denote the @xmath97-shadow corresponding to  @xmath68 . then  @xmath109\!]}^{\mathrm{n}}=0 $ ] iff  @xmath110 . similarly if  @xmath111 and  @xmath112 are bivectors then  @xmath56\!]}^{\mathrm{s}}=0 $ ] iff  @xmath113 where  @xmath114 denotes the @xmath98-shadow corresponding to  @xmath52 . in both cases the curly brackets denote the _ super _ bracket of shadows that arises due to oddness of the variables  @xmath93 and  @xmath96 . additional discussion of theorem  [ thm : brackets ] the reader will find in remark  [ rem : nonlocal ] . [ thm : correpondence ] to any cosymmetry of  @xmath22 there canonically corresponds a conservation law of  @xmath115 . dually to any symmetry of  @xmath22 there canonically corresponds a conservation law of  @xmath116 . let locally the equation  @xmath22 be given by the system @xmath117 where  @xmath118 and  @xmath119 . [ [ sec : step-1 ] ] step 1 + + + + + +    consists of writing out defining equations for symmetries and cosymmetries of  @xmath22 . let  @xmath120 be _ internal coordinates _ on  @xmath22 , @xmath121 and  @xmath122 being some sets of ( multi)indices and  @xmath123 corresponding to  @xmath124 . then any @xmath9-field on  @xmath22 is a linear combination of the _ total derivatives _ @xmath125 the linearization of  @xmath22 is the matrix operator with the entries @xmath126 a symmetry  @xmath127 enjoys the equation @xmath128 and the corresponding field is the _ evolutionary vector field _ @xmath129 while the bracket of symmetries  @xmath30 , @xmath90 is given by @xmath130 the operator adjoint to   is @xmath131 and a cosymmetry  @xmath132 satisfies the equation @xmath133    [ [ sec : step-2 ] ] step 2 . + + + + + + +    here we look for closed @xmath7-forms and construct abelian coverings associated to them . a horizontal form  @xmath134 is closed if @xmath135 such a form gives rise to a nonlocal variable  @xmath136 that satisfies the equations @xmath137 these equations are compatible on   due to  . recall that for  @xmath138 closed @xmath7-forms coincide with conservation laws . the total derivatives lifted to the covering equation  @xmath71 are @xmath139    [ [ sec : step-3 ] ] step 3 . + + + + + + +    at this step we compute a number of particular symmetries and cosymmetries ( using equations   and  , resp . ) . they are used to construct canonical nonlocal variables on the @xmath6-covering ( _ nonlocal vectors _ ) and on the @xmath5-covering ( _ nonlocal forms _ ) , resp . , at step  4 . we also use them as seed elements in series generated by recursion operators . [ [ sec : step-4 ] ] step 4 + + + + + +    consists of construction of the @xmath5- and @xmath6-coverings and introduction of canonical nonlocal variables over them ( see step  3 ) . the @xmath5-covering is obtained by adding to   the system of equations @xmath140 cf .  with eq . , while the @xmath6-covering is given by @xmath141 that comes from  . if  @xmath142 is a symmetry of  @xmath22 then one can introduce a covering over  @xmath116 described by the system @xmath143 where  @xmath144 are @xmath9-differential operators ( see theorem  [ thm : correpondence ] ) . in a similar way , to any cosymmetry  @xmath145 there corresponds a covering @xmath146 @xmath147 being @xmath9-differential operators as well . we omit here a general description of these operators and refer the reader to the particular case of our interest exposed in sections  [ sec : nonlocal - forms ] and  [ sec : nonlocal - vectors ] . [ [ sec : step-5 ] ] step 5 . + + + + + + +    we now use theorems  [ thm : ell - shad ] and  [ thm : ellstar - shad ] to construct recursion operators and hamiltonian and symplectic structures . let  @xmath148 cosymmetries of  @xmath22 . let us consider the covering  @xmath149 over  @xmath115 with the nonlocal variables  @xmath150 defined by   and lift the operators  @xmath37 and  @xmath151 to this covering . then the following result specifies theorem  [ thm : ell - shad ] :    [ thm : nloc - ell ] let  @xmath152 be a solution of the equation  @xmath153 on  @xmath149 linear w.r.t . @xmath154 and @xmath155 @xmath156 then the operator @xmath157 takes shadow of symmetries to shadows of symmetries . in a similar way to any solution  @xmath158 @xmath159 there corresponds the operator @xmath160 that takes shadows of symmetries to shadows of cosymmetries .    in a dual way , consider symmetries  @xmath161 of the equation  @xmath22 and the covering  @xmath162 over  @xmath116 with the nonlocal variables  @xmath163 defined by  . then , lifting  @xmath37 and  @xmath151 , we obtain a similar specification of theorem  [ thm : ellstar - shad ] :    [ thm : nloc - ellstar ] let  @xmath152 be a solution of the equation  @xmath153 on  @xmath162 linear w.r.t . @xmath164 and @xmath165 @xmath166 then the operator @xmath167 takes shadow of cosymmetries to shadows of symmetries . in a similar way to any solution  @xmath158 @xmath168 there corresponds the operator @xmath169 that takes shadows of cosymmetries to shadows of cosymmetries .    after finding the operators  @xmath68 , @xmath170 , @xmath52 and  @xmath171 we check conditions   and   and compute necessary schouten and nijenhuis brackets .    [ [ sec : step-6 ] ] step 6 + + + + + + the last step consists of establishing algebraic relations between the invariants constructed above . we consider eq .   in the form @xmath172 i.e. , set  @xmath173 , and , similar to  , introduce a new variable  @xmath174 , where  @xmath175 is a new real constant . consequently , the initial equation transforms to the system @xmath176 we choose the following variables for internal local coordinates on the infinite prolongation of eq .  : @xmath177 then the total derivatives in these coordinates will be of the form @xmath178    we introduce the following gradings : @xmath179 and extend them in a natural way to all polynomial functions of the internal coordinates . then all computations can be restricted to homogeneous components .      in subsequent computations we shall need the following nonlocal variables arising from conservation laws and defined by the equations @xmath180 the variable  @xmath181 is of grading  @xmath34 and computational experiment shows that for every grading  @xmath182 , @xmath183 , there exist an  @xmath181 such that  @xmath184 . in addition , we found conservation laws of fractional gradings : @xmath185 etc . a symmetry  @xmath186 of eq .   must satisfy the linearized equation @xmath187 direct computations lead to the following results . [ [ sec : x - t - independent ] ] ( x , t)-independent symmetries . + + + + + + + + + + + + + + + + + + + + + + + + + + + + +    one can observe two types of symmetries that are independent of  @xmath188 and  @xmath189 . the first one consists of symmetries of integer gradings : @xmath190 etc . symmetries of the second type have semi - integer gradings : @xmath191 etc . all symmetries are local and  @xmath192 . if one adds to the nonlocal setting the variables  @xmath193 ( see above ) then an additional series of nonlocal symmetries arises : @xmath194 etc . [ [ sec : x - t - dependent ] ] ( x , t)-dependent symmetries . + + + + + + + + + + + + + + + + + + + + + + + + + + +    the first three symmetries that depend on  @xmath188 and  @xmath189 are @xmath195 all these symmetries , except for the first one , are nonlocal ( description of the nonlocal variable is given in subsection  [ sec : nonlocal - variables-1 ] ) and , as above , the subscript denotes the grading . the defining equation for cosymmetries  @xmath196 is the adjoint to the linearization of  : @xmath197 similar to symmetries , we consider two types of cosymmetries .    [ [ sec : x - t - independent-1 ] ] ( x , t)-independent cosymmetries . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +    they are local and may be of integer and semi - integer gradings : @xmath198 etc . similar to the case of symmetries , when one adds nonlocal variables  @xmath193 an additional series of nonlocal cosymmetries arises : @xmath199 etc . [ [ sec : x - t - dependent-1 ] ] ( x , t)-dependent cosymmetries . + + + + + + + + + + + + + + + + + + + + + + + + + + + + +    all them are nonlocal : @xmath200 etc . recall that nonlocal forms are nonlocal variables of a special type on the @xmath5-covering . the @xmath5-covering itself is obtained from eq .   by adding two additional equations @xmath201 where  @xmath202 and  @xmath203 are new odd variables . the total derivatives on the @xmath5-covering are @xmath204 the nonlocal form  @xmath205 associated to a cosymmetry  @xmath206 ( see subsection  [ sec : cosymmetries-1 ] ) is defined by the equations @xmath207      dually to nonlocal forms , nonlocal vectors arise as special nonlocal variables on the @xmath6-covering associated to symmetries of the initial equation . the @xmath6-covering is the extension of eq .   by two new equations @xmath208 where  @xmath209 is a new odd variable . the total deriavatives are given by @xmath210 the nonlocal vector  @xmath211 associated to a symmetry  @xmath186 ( see subsection  [ sec : symmetries-1 ] ) is defined by the equations @xmath212      the defining equations for these operators are @xmath213 ( see theorem  [ thm : nloc - ell ] ) , where the total derivatives are those described in subsection  [ sec : nonlocal - forms-1 ] . the following two solutions are essential : @xmath214 the corresponding operators are of the form @xmath215 where  @xmath216 , @xmath217 . all other solutions obtained in our computations corresponded to operators that are generated by the two above . symplectic structures , as it follows from theorem  [ thm : nloc - ell ] , are defined by the equations @xmath218 where the total derivatives were defined in subsection  [ sec : nonlocal - forms-1 ] . here are the simplest nontrivial solutions : @xmath219 with the corresponding symplectic operators @xmath220      the equations that should be satisfied by a hamiltonian operator ( see theorem  [ thm : nloc - ellstar ] ) are @xmath221 where the total derivatives are from subsection  [ sec : nonlocal - vectors-1 ] . in particular , we found the following solutions : @xmath222 the corresponding hamiltonian operators are @xmath223      by theorem  [ thm : nloc - ellstar ] , the equation to find recursion operators for cosymmetries are @xmath224 with the total derivatives given in subsection  [ sec : nonlocal - vectors-1 ] . one of solutions is presented below : @xmath225 the corresponding recursion operator is @xmath226      we expose here basic facts on structural relations between the above described invariants . the main one is the following    [ thm : p - n ] recursion operator  @xmath227 and hamiltonian operator  @xmath228 constitute a poisson  nijenhuis structure on the camassa  holm equation . consequently all the operators  @xmath229 are hamiltonian and pair - wise compatible . in particular @xmath230 . the proof consists of direct computations using the results and techniques of ref . @xcite is also a poisson  nijenhuis structure and generates an infinite negative hierarchy of hamiltonian operators , but could not prove this fact because of computer capacity limitations . ] . a visual presentation of how symmetries are distributed over gradings is given in table  [ tab:2 ] . l|ccccccccccccccccc gradings & @xmath231 & @xmath232 & @xmath233 & @xmath234 & @xmath235 & @xmath236&@xmath7&@xmath58&@xmath237&@xmath238&@xmath239&@xmath240&@xmath241&@xmath242 + local positive & & & & & & & @xmath243 & @xmath243 & & & @xmath243&@xmath243 + local negative & @xmath243 & & @xmath243 & & & & & & + nonlocal positive & & & & & & @xmath243 & & & @xmath243 & @xmath243 & & & @xmath243 & @xmath243 + nonlocal negative & & @xmath243 & & @xmath243 & & & & & +    how to prove locality of the first two series of symmetries will be discussed in section  [ sec : conclusion ] . similar presentation for cosymmetries see in table  [ tab:3 ] . l|ccccccccccccccccc gradings & @xmath233 & @xmath234 & @xmath235 & @xmath236 & @xmath244 & @xmath7&@xmath245&@xmath58&@xmath237&@xmath238&@xmath239&@xmath240&@xmath241&@xmath242 + local positive & & & & & & & & & @xmath243 & @xmath243&&&@xmath243&@xmath243 + local negative & @xmath243 & & @xmath243 & & @xmath243 & & @xmath243 & & + nonlocal positive & & & & & & & & @xmath243 & & & @xmath243 & @xmath243 & & + nonlocal negative & & @xmath243 & & @xmath243 & & @xmath243 & & & +    the action of hamiltonian and recursion operators for symmetries ( up to a constant multiplier ) is given in diagram  : @xmath246_{\tens{r}_3}&\phi_{-5/2}\ar[r]_{\tens{r}_3}\ar@/{_}/[l]_{\tens{r}_{-1 } } &      \phi_{-3/2}\ar[r]_{\tens{r}_3}\ar@/{_}/[l]_{\tens{r}_{-1}}&0 &      \phi_1\ar[r]_{\tens{r}_3}\ar@/{_}/[l]_{\tens{r}_{-1 } } &      \phi_2\ar[r]_{\tens{r}_3}\ar@/{_}/[l]_{\tens{r}_{-1 } }      & \dots\ar@/{_}/[l]_{\tens{r}_{-1}}\\      \dots&\psi_{1/2}\ar[ur]^{\tens{h}_{-2}}\ar[u]^(.3){\tens{h}_{-3 } } &      \psi_{3/2}\ar[ur]^{\tens{h}_{-2}}\ar[u]^(.3){\tens{h}_{-3 } } &      \psi_3\ar[ur]^{\tens{h}_{-2}}\ar[u]^(.3){\tens{h}_{-3 } } &      \psi_4\ar[ur]^{\tens{h}_{-2}}\ar[u]^(.3){\tens{h}_{-3 } } &      \psi_7\ar[ur]^{\tens{h}_{-2}}\ar[u]^(.3){\tens{h}_{-3 } }      & \dots    } \ ] ] the action of recursion operators for cosymmetries and simplectic structures is similar . we shall now prove commutativity of the local hierarchies . [ lem : action ] the symmetry  @xmath247 is a positive hereditary symmetry i.e. its action on local symmetries @xmath248 coincides up to a multiplier with the one of the recursion operator  @xmath227 . the only symmetries that vanish under this action are  @xmath249 and  @xmath30 . in a similar way the symmetry @xmath250 is a negative hereditary symmetry and the only symmetry that is taken to zero under its action is  @xmath30 . a direct corollary of this result is    [ thm : commut ] local positive and negative symmetries form commutative hierarchies . let us consider now the camassa holm equation in its initial form   with  @xmath173 and , similar to the matrix case introduce a new real parameter  @xmath175 : @xmath251 for the internal coordinates we choose the functions @xmath252 the total derivatives in these coordinates are of the form @xmath253 where  @xmath254 . the equation becomes homogeneous if assign the following gradings : @xmath255      we introduce nonlocal variable associated to conservation laws of the equation at hand : @xmath256      a symmetry  @xmath142 must satisfy the linearized equation @xmath257    we computed two types of symmetries . everywhere below the subscript was chosen in a way to correspond the enumeration taken for the matrix case . [ [ sec : x - t - independent-2 ] ] @xmath258-independent symmetries . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +    we present the first four of them : @xmath259 all these symmetries are local . [ [ sec : x - t - dependent-2 ] ] @xmath258-dependent symmetries . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +    the first three @xmath258-dependent symmetries are @xmath260 the only local symmetry in this series is  @xmath30 . the defining equation for cosymmetries is @xmath261    [ [ sec : x - t - independent-3 ] ] @xmath258-independent cosymmetries . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +    these cosymmetries are local . the first four of them are @xmath262    [ [ sec : x - t - dependent-3 ] ] @xmath258-dependent cosymmetries . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +    these cosymmetries are nonlocal , except for the first one : @xmath263 etc . nonlocal forms arise in the @xmath5-covering which is given by the equation @xmath264 with the total derivatives @xmath265 where  @xmath266 .    to any cosymmetry  @xmath267 we associate a nonlocal form  @xmath268 defined by the equations @xmath269      nonlocal vectors arise in the @xmath6-covering . the latter is the extension of the initial equation by the equation @xmath270 with the total derivatives @xmath271 where @xmath272 .    to any symmetry  @xmath273 there corresponds a nonlocal vector  @xmath274 defined by the relations @xmath275      the defining equation for these operators is @xmath276 where the total derivatives are those presented in subsection  [ sec : nonlocal - forms ] . we consider here two nontrivial solutions : @xmath277 the operator corresponding to the second one is @xmath278 the first operator is too complicated to present it here . the equation defining symplectic structures is @xmath279 where the total derivatives are from subsection  [ sec : nonlocal - forms ] . the simplest solutions are @xmath280 while the corresponding operators have the form @xmath281      hamiltonian structures are defined by the equation @xmath282 where the total derivatives are given in subsection  [ sec : nonlocal - vectors ] . we present here three solutions . two of them are local and the third one is nonlocal : @xmath283 the corresponding operators are @xmath284      these recursion operators are defined by the equation @xmath285 where the total derivatives are given in subsection  [ sec : nonlocal - vectors ] . one of the solutions is @xmath286 to which the operator @xmath287 corresponds . we finish this paper with a number of remarks . [ rem : what - has - been ] first of all , let us stress again what has been done above . we treated the camassa holm equation directly , without artificial assumptions about its evolution or pseudo - evolution nature . the pass from the original form   to system   ( which is _ not _ in evolution form ) was done by technical reasons only . on this way , we found an infinite family of pair - wise compatible hamiltonian structures , recursion operators for symmetries and cosymmetries , and symplectic operators . these structures lead to existence of two commutative series of local symmetries and conservation laws and thus the equation is integrable . [ rem : nonlocal ] several comments are worth to be made in relation to theorem  [ thm : brackets ] . two cases must be distinguished : local and nonlocal . in both cases the problem reduces to _ reconstruction _ of nonlocal shadows up to symmetries and subsequent computation of the bracket . this case is simple and the result is as follows : any shadow in the @xmath5-covering corresponding to a recursion operator can be canonically lifted up to a symmetry of the covering equation and the commutator of these symmetries corresponds to the nijenhuis bracket of the operators . a similar result is valid for the shadows in the @xmath6-covering corresponding to local bivectors ( in this case the commutator of symmetries is related to the schouten bracket ) . the case of nonlocal operators is much more complicated and rests on the problem of how to commute nonlocal shadows . seemingly there exists no _ natural _ definition of such a commutator , but in ref . @xcite we proposed a procedure both to reconstruct and commute shadows . a weak point of this procedure is that it is based on rather cumbersome and intuitively not obvious notion of shadow equivalence . dealing with equivalence classes of shadows necessitates elimination of certain kind of nonlocal variables that we call _ pseudo - constants _ and that are intrinsically related to the covering at hand . this not always simple to do and we plan to simplify and clarify the procedure . [ rem : comp - two - repr ] of course , all invariants of the camassa  holm equation in its initial form can be obtained from those computed for the system by simple transformations . e.g. , to obtain symmetries of eq . from those of   one should substitute in the @xmath288-component the variables @xmath289 by @xmath290 . on the other way , any cosymmetry in the scalar case can be constructed directly from the corresponding cosymmetry of eq .   by changing the variables  @xmath289 in the @xmath291-component by  @xmath290 . similar transformations are applicable to other structures . the following correspondence between the components of symmetries and cosymmetries of matrix equation   is valid @xmath292^{d_t+ud_x - u_1}&\psi_k^u\ar@{|-{>}}[r]^{\cdot(-1 ) } &          \phi_{k-2}^u\ar@{|-{>}}[r]^{\alpha - d_x^2}&\phi_{k-2}^w .        } \ ] ]    [ rem : local - ov - hier ] now we shall show how to prove locality of symmetry hierarchies . let us introduce a new variable  @xmath293 . then eq .   transforms to @xmath294 i.e. , the first equation acquires potential form , see above . ] . this means that for any symmetry  @xmath295 the form  @xmath296 is a conservation law of eq .  . comparing gradings , it is easily checked that all conservation laws of the form  @xmath297 are trivial and consequently  @xmath298 lies in the image of  @xmath299 for any symmetry  @xmath142 . from this fact it follows that the action of recursion operators is local . [ sec : relations - n - m ] finally , we indicate relations between the camassa  holm equation and the equation describing short capillary - gravity waves ( the neveu  manna equation , see  @xcite ) @xmath300 we strongly believe that there exists a deformation connecting these two equations and their integrable structures are closely related to each other . we intend to discuss this relation elsewhere . aratyn h. , gomes j.f . , zimerman a.h . , on a negative flow of the akns hierarchy and its relation to a two - component camassa holm equation , sigma ( symmetry , integrability and geometry : methods and applications ) , * 2 * , 070 ( 2006 ) , arxiv : hep - th/0610182 . bocharov , a.v . , chetverikov , v.n . , duzhin , s.v . , et al . , symmetries and conservation laws for differential equations of mathematical physics , xiv+333  pp . amer . soc . , providence , ri ( 1999 ) . edited and with a preface by i.  krasilshchik and a.  vinogradov . igonin s. , verbovetsky a. , vitolo r. , variational multivectors and brackets in the geometry of jet spaces , v int . conf .  on on symmetry in nonlinear mathematical physics , kyiv 2003 inst .  of mathematics , acad .  of sciences of ukraine , pp . 1335 - 1342 . see also http://www.math.utwente.nl/publicaties/2002/1641abs.html . kersten p. , krasilshchik i. , verbovetsky a. , ( non)local hamiltonian and symplectic structures , recursions , and hierarchies : a new approach and applications to the @xmath302 supersymmetric kdv equation , j.  phys . , * 37 * , pp . 50035019 ( 2004 ) , arxiv : nlin/0305026 .
using geometrical approach exposed in refs . 
 @xcite , we explore the camassa  holm equation ( both in its initial scalar form , and in the form of @xmath0-system ) . 
 we describe hamiltonian and symplectic structures , recursion operators and infinite series of symmetries and conservation laws ( local and nonlocal ) .