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the problem of describing all finite - dimensional lie algebras of vector fields is a starting point for the symmetry analysis of ordinary differential equations , because , having solved this problem , one finds all possible algebras of contact symmetries for ordinary differential equations .    over the complex numbers this classification was done at the end of the last century by sophus lie  @xcite . he showed that , with three exceptions , all lie algebras of contact vector fields , viewed up to equivalence , are lifts of lie algebras of vector fields on the plane . the largest algebra of the three exceptions ( so - called _ irreducible algebras of contact vector fields _ ) is the algebra of contact symmetries of the equation @xmath0 and is isomorphic to @xmath1 , while the other two are its subalgebras of dimension 6 and  7 .    in this paper we show that the problem of describing algebras of vector fields can be formulated in a natural way in terms of filtered and graded lie algebras . this allows not only to give a new up - to - date proof of sophus lie s classification , which is as yet missing in the literature , but also to solve this problem over the field of real numbers . it turns out that in the real case there are 8  irreducible contact lie algebras of vector fields on the plane , and one of them involves an arbitrary parameter . lie algebras of vector fields on the plane were also classified ( both in real and complex case ) by sophus lie  @xcite , so that the description of irreducible lie algebras of vector fields on the plane which is given in the present paper , basically concludes the description of all finite - dimensional contact lie algebras of vector fields over the field of real numbers . it should be noted that the problem of finding all irreducible contact lie algebras over the real numbers was also considered by f.  engel in  @xcite , which is mentioned in sophus lie s three - volume treatise  @xcite ( volume 3 , chapter  29 , pages  760761 ) . p  olver , in his recently published book  @xcite , cites this problem as unsolved . let @xmath2 be the set of 1jets of mappings from  @xmath3 to  @xmath3 , and let  @xmath4 denote the natural projection @xmath5 . we fix a coordinate system @xmath6 on  @xmath7 in which the 1jet of the mapping @xmath8 at the point  @xmath9 has the coordinates @xmath10 . the projection  @xmath4 has in these coordinates the form @xmath11 . we can introduce a natural _ contact structure _ indeed , there is a two - dimensional distribution  @xmath12 on  @xmath7 which is not completely integrable and has the property that all its integral curves whose projection onto the plane is diffeomorphic , are precisely the curves of the form @xmath13 with @xmath14 , @xmath15 . in terms of coordinates , this distribution is given by the vector fields  @xmath16 and @xmath17 or , alternatively , by the differential 1form @xmath18 . a ( local ) diffeomorphism  @xmath19 of the manifold  @xmath7 is said to be _ contact _ if @xmath19  preserves the contact distribution  @xmath12 , ie , if @xmath20 for all @xmath21 . a vector field on  @xmath7 is called contact if it generates a local one - parameter transformation group that consists solely of contact diffeomorphisms . it is easy to show that a vector field  @xmath22 is contact if and only if @xmath23 for some smooth function  @xmath24 . if @xmath22 is a contact vector field , then the function @xmath25 is called the _ characteristic function _ of  @xmath22 . it completely determines the field  @xmath22 , which in this case is denoted by  @xmath26 and has the form @xmath27    the mapping @xmath28 establishes an isomorphism between the space of all smooth functions and that of contact vector fields on  @xmath7 . this allows to make the space @xmath29 into a lie algebra by letting @xmath30)$ ] . if @xmath19 is a ( local ) diffeomorphism of the plane , then there exists a unique local contact transformation @xmath31 such that the following diagram is commutative : @xmath32 the transformation  @xmath33 is then called the ( _ first _ ) _ prolongation of the diffeomorphism _  @xmath19 and , in terms of coordinates , has the from @xmath34 similarly , for any vector field  @xmath22 on the plane there exists a unique contact vector field  @xmath35 on @xmath36 such that @xmath37 . this vector field  @xmath35 is called the ( _ first _ ) _ prolongation of the vector field _ @xmath22 and has the form @xmath38 its characteristic function is @xmath39 . the mapping @xmath40 is an embedding of the lie algebra of vector field on the plane into the lie algebra of contact vector fields on  @xmath36 . the contact vector fields that lie in the image of this mapping are called _ point _ contact vector fields . point vector fields  @xmath41 are characterized by the following two equivalent properties :    1 . any point vector field  @xmath41 is an infinitesimal symmetry of the _ vertical distribution _ @xmath42 on  @xmath36 ( @xmath43 ) ; 2 . the characteristic function of  @xmath41 is linear in  @xmath44 . a lie algebra  @xmath45 of contact vector fields is called _ reducible _ if there is a local contact diffeomorphism  @xmath19 such that the lie algebra @xmath46 consists only of point vector fields . otherwise , @xmath45  is said to be _ irreducible_.    [ th1 ] a lie algebra  @xmath45 of contact vector fields is irreducible if and only if it preserves no one - dimensional subdistribution of the contact distribution . every lie algebra that consists of point vector fields preserves the vertical distribution  @xmath42 , which is a one - dimensional subdistribution of the contact distribution  @xmath12 . consequently , any reducible lie algebra of contact vector fields also preserves a one - dimensional subdistribution of  @xmath12 . conversely , let @xmath45 be a lie algebra of vector fields that preserves some one - dimensional subdistribution  @xmath47 of the contact distribution . if @xmath48 and  @xmath49 are two functionally independent first integrals of  @xmath47 , then , as one can easily verify , the local diffeomorphism @xmath50 is contact and transforms the vertical distribution  @xmath42 to  @xmath47 . it follows that the lie algebra @xmath51 preserves the vertical distribution and hence consists of point vector fields . any irreducible lie algebra of contact vector fields is transitive at a point in general position . let @xmath45 be an irreducible lie algebra of contact vector fields . for an arbitrary point @xmath52 , we let @xmath53 and define @xmath54 and @xmath55 . then @xmath56 is obviously an open subset in  @xmath36 . the lie algebra  @xmath45 is transitive at point in general position if and only if @xmath57 . assume the contrary . then the subspaces  @xmath58 form a completely integrable distribution  @xmath47 in  @xmath56 which is invariant under  @xmath45 . consider the following two possibilities :    1@xmath59 : @xmath60 then the intersection @xmath61 is one - dimensional at the points in general position , and this family of subspaces forms a one - dimensional subdistribution of the contact distribution which is invariant under  @xmath45 .    2@xmath59 : @xmath62 in this case @xmath47  can be locally embedded into a two - dimensional completely integrable distribution which , as follows from its construction , is also invariant under  @xmath45 . then , arguing as in the previous case , we conclude that the lie algebra  @xmath45 preserves a one - dimensional subdistribution of the contact distribution .    in this paper we restrict ourselves to a local description of finite - dimensional lie algebras of contact vector fields at a point in general position . in particular , from now on we shall assume that all irreducible algebras of contact vector fields are transitive . let @xmath45 be a transitive lie algebra of contact vector fields on  @xmath2 , let @xmath63  be an arbitrary point in  @xmath36 , and let @xmath64 be the subalgebra of  @xmath45 that consists of all vector fields in  @xmath45 vanishing at the point  @xmath63 . it is easy to show that the subalgebra  @xmath65 is _ effective _ , ie , contains no nonzero ideals of  @xmath45 ( see , for example , ( * ? ? ? * theorem 10.1 ) ) . we can identify @xmath66 with @xmath67 in the obvious way . then @xmath68 is identified with a certain submodule  @xmath69 of the @xmath65module @xmath67 . since the distribution  @xmath12 is not completely integrable , the subspace @xmath70 will not be closed with respect to the multiplication in  @xmath45 . we define a decreasing chain of subspace in  @xmath45 as follows : @xmath71 for all @xmath72 , @xmath73\subset { \mathfrak g}_p\}\text { for all } p\ge0.\ ] ] it is easily shown by induction that @xmath74\subset{\mathfrak g}_{p+q}$ ] for all @xmath75 , so that the family of subspaces @xmath76 defines a filtration of the lie algebra  @xmath45 . a filtered lie algebra  @xmath45 is called a _ contact lie algebra _ if    a.   @xmath71 for all @xmath72 ; b.   @xmath77 , @xmath78 , and @xmath79+{\mathfrak g}_{-1}={\mathfrak g}$ ] ; c.   @xmath80\subset { \mathfrak g}_p\}$ ] for all @xmath81 ; d.   @xmath82 .    two contact lie algebras are said to be isomorphic if they are isomorphic as filtered lie algebras . show that any transitive lie algebra  @xmath45 of contact vector fields is a contact lie algebra with respect to the above filtration . the properties a ) and c ) follow immediately from the way that the filtration in  @xmath45 is introduced . let us prove b ) . since the contact distribution has codimension  1 , we get @xmath77 . from transitivity of @xmath45 on @xmath7 we get @xmath83 . next , since @xmath12 is not completely integrable , the subspace @xmath79+{\mathfrak g}_{-1}$ ] is strictly greater than @xmath84 and , hence , is equal to @xmath45 . finally , let @xmath85 . then , obviously , @xmath86 is an ideal in  @xmath45 contained in  @xmath65 . now since @xmath65 is an effective subalgebra , it follows that @xmath87 . this proves  d ) . conversely , let @xmath45 be an arbitrary finite - dimensional contact lie algebra . then the pair @xmath88 determines a unique ( up to local equivalence ) realization of  @xmath45 as a transitive lie algebra of vector fields on  @xmath89 . and the subspace  @xmath84 allows us to define a @xmath45invariant two - dimensional distribution on  @xmath89 which is not completely integrable . therefore , the lie algebra  @xmath45 admits a unique ( up to local equivalence ) realization as a transitive lie algebra of contact vector fields . thus , the local classification of finite - dimensional transitive lie algebras of contact vector fields on  @xmath36 is equivalent to the classification ( up to isomorphism ) of the corresponding contact lie algebras . observe that the latter problem is algebraic and , as we shall see later , can be solved by purely algebraic means . all @xmath45invariant distributions on  @xmath36 are in one - to - one correspondence with the submodules of the @xmath65module @xmath67 . in particular , the contact distribution corresponds to the submodule @xmath90 . from theorem  [ th1 ] now easily follows the next algebraic criterion for the irreducibility of  @xmath45 .    [ lem1 ] a transitive lie algebra  @xmath45 of contact vector fields is irreducible if and only if the @xmath65module @xmath90 is irreducible . the major tool in the study of filtered lie algebras is to consider graded lie algebras associated with them . as we shall see later on , with a few exceptions , irreducible contact lie algebras can be completely restored from their associated graded lie algebras .    a @xmath91graded lie algebra @xmath92 is called a _ graded contact lie algebra _ if    a.   @xmath93 for all @xmath94 ; b.   @xmath95 , @xmath96 , and @xmath97={\mathfrak h}_{-2}$ ] ; c.   @xmath98=0\}=\{0\}$ ] for all @xmath81 . if @xmath45 is a contact lie algebra , then it is clear that the associated graded lie algebra @xmath99 satisfies all three conditions in the above definition and is therefore a graded contact lie algebra . moreover , if @xmath45 is a filtered lie algebra such that the associated graded lie algebra  @xmath100 is contact and @xmath101 , then it is easy to show that @xmath45 itself is a contact lie algebra . the concept of irreducibility for contact lie algebras can be carried over to graded contact lie algebras . from lemma  [ lem1 ] it immediately follows that a contact lie algebra  @xmath45 is irreducible if and only if so is the @xmath102module @xmath103 in the corresponding graded contact lie algebra  @xmath100 . the graded contact lie algebras that satisfy this condition will be called _ irreducible_.    the classification of all irreducible graded contact lie algebras can be carried out using the methods developed in the works of tanaka  @xcite . slightly modifying the terminology of those papers ( see also  @xcite ) , we introduce the concept of transitive graded lie algebra , which generalizes the concept of graded contact lie algebra . a graded lie algebra @xmath104 is said to be _ transitive _ if it satisfies the following conditions :    a.   there exists a natural number @xmath105 such that @xmath106 for all @xmath107 ; b.   @xmath108={\mathfrak g}_{-p-1}$ ] for all @xmath109 ; c.   if @xmath110 for @xmath111 and @xmath112=\{0\}$ ] , then @xmath113 .    it immediately follows from this definition that @xmath114 is the graded nilpotent lie algebra generated by  @xmath84 . following tanaka  @xcite , we shall call graded nilpotent lie algebras of this kind _ fundamental_. in particular , the fundamental nilpotent lie algebra corresponding to a graded contact lie algebra is none other than the three - dimensional heisenberg algebra . let @xmath115 be an arbitrary fundamental graded lie algebra . then , as was shown by tanaka  @xcite , there exists a unique transitive graded lie algebra @xmath116 that satisfies the following conditions :    1 . @xmath117 for @xmath118 ; 2 . @xmath119 is the largest among all transitive graded lie algebras satisfying condition  ( 1 ) . this lie algebra  @xmath119 is called the ( _ algebraic _ ) _ extension _ of  @xmath45 . in particular , any transitive graded lie algebra  @xmath45 may be identified with a graded subalgebra of  @xmath119 , where @xmath115 . the lie algebra  @xmath119 has a clear geometrical meaning . namely , let @xmath7  be a connected lie group with lie algebra  @xmath120 , and let  @xmath121 be a left - invariant distribution on  @xmath7 such that @xmath122 . denote by  @xmath123 the lie algebra of all germs of infinitesimal symmetries of  @xmath121 at the identity element  @xmath124 of  @xmath7 . consider the following two subspaces in  @xmath123 : @xmath125 where @xmath126 denotes the germ of the vector field  @xmath22 at the point  @xmath124 . now let @xmath127\text { for all } p\geq1\\    a_p=&\{\xi\in{\mathcal a}_{p-1}\mid[\xi,{\mathcal a}_{-1}]\subset{\mathcal a}_{p-1}\}\text { for all }    p\geq1.\end{aligned}\ ] ] then the family of subspaces @xmath128 forms a decreasing filtration of the lie algebra  @xmath123 , and @xmath119 can be identified with the associated graded algebra , ie , @xmath129 for all @xmath130 . this geometrical interpretation allows to describe , without difficulty , the structure of  @xmath119 in the case that we are interested in , namely in the case of graded contact lie algebras . let @xmath131 be the three - dimensional heisenberg algebra and @xmath132 $ ] , while @xmath133 is a two - dimensional subspace complementary to  @xmath134 $ ] . in this case we may assume without loss of generality that @xmath121  is precisely the contact distribution on  @xmath36 .    using the description of all infinitesimal symmetries of the contact distribution , it is not hard to determine the structure of the lie algebra  @xmath135 . it can be identified with the space of polynomials in @xmath136 with the bracket operation given by @xmath137 $ ] . the space  @xmath138 consists of all homogeneous polynomials of degree  @xmath139 , assuming that the variables @xmath136 are of degree 1 , 2 and  1 respectively . for example , @xmath140    we shall now fix some fundamental graded lie algebra  @xmath120 and describe how one can classify all finite - dimensional graded subalgebras  @xmath100 of the lie algebra  @xmath119 such that @xmath141 for all @xmath111 .    in what follows we shall always assume that @xmath142 for all @xmath118 . suppose that for some @xmath143 we have a collection of subspaces @xmath144 , @xmath145 , such that @xmath146\subset{\mathfrak h}_{p+q}$ ] @xmath147 , @xmath148 . using induction , we define a sequence of subspaces @xmath149 as follows : @xmath150\subset{\mathfrak h}_p\}\ ] ] for all @xmath151 . it can be easily shown that @xmath152 is a graded subalgebra of  @xmath119 . this subalgebra is called the _ extension _ of the collection @xmath153 . note that @xmath154 is the largest of all graded subalgebras whose @xmath155th grading space coincides with  @xmath156 for all @xmath157 . one the other hand , we can associate @xmath153 with the graded subalgebra @xmath158 generated by these subspaces . now let @xmath100 be an arbitrary graded subalgebra of  @xmath119 such that @xmath159 for all @xmath160 . then , obviously , for any @xmath161 we have @xmath162 finally , notice that @xmath163 is precisely the algebra of all derivations of  @xmath120 that preserve the grading ( see  @xcite ) , and all the subspaces @xmath164 are invariant under the natural action of  @xmath102 on  @xmath165 . based on these remarks , the following algorithm for the classification of the desired kind of subalgebras in  @xmath119 suggests itself .    * * step  i**describe , up to conjugation , all subalgebras @xmath166 . go to step  iii .    * * step  ii**suppose that for some @xmath143 , a collection of subspaces @xmath144 , @xmath145 , is already constructed such that @xmath167\subset{\mathfrak h}_{p+q}\quad\forall p , q\leq k,\ p+q\leq k\\ { \rm(ii)}&\quad\dim\tilde{\mathfrak g}({\mathfrak m},{\mathfrak h}_0,\dots,{\mathfrak h}_k)<\infty.\end{aligned}\ ] ] let @xmath168,\\ & { \mathfrak g}_{k+1}({\mathfrak m},{\mathfrak h}_0,\dots,{\mathfrak h}_k)=\{x\in{\mathfrak g}_{k+1}({\mathfrak m})\mid[x,{\mathfrak h}_{-1}]\subset{\mathfrak h}_k\}.\end{aligned}\ ] ] at this point we describe all @xmath102-invariant subspaces  @xmath169 in @xmath170 such that @xmath171    * * step iii**find the subalgebras @xmath172 and @xmath173 . if these subalgebras are not the same , go to step  ii . if , however , they coincide , then @xmath174 is one of the desired subalgebras . now we shall use this algorithm to classify all irreducible graded contact lie algebras over the field of real numbers . [ thm1 ] let @xmath131 denote the three - dimensional real heisenberg algebra , considered as a graded lie algebra , and let @xmath135 be the universal extension of  @xmath131 . then any finite - dimensional irreducible graded contact lie algebra  @xmath100 is isomorphic to one and only one of the following subalgebras of  @xmath135 : @xmath175 2 . @xmath176 3 . @xmath177 4 . @xmath178 5 . @xmath179 6 . @xmath180 , @xmath181    fix a basis @xmath182 in the space @xmath183 . then the action of the elements of  @xmath184 on @xmath183 is given by the following matrices : @xmath185 therefore , the lie algebra  @xmath184 may be identified with @xmath186 , and the @xmath184module @xmath183 with the natural @xmath186module . [ lem2 ] any irreducible subalgebra of  @xmath186 is conjugate to one and only one of the following subalgebras : @xmath187    { \rm(iii)}&\ { \mathfrak{sl}}(2,{\mathbb r})&{\rm(iv)}&\ { \mathfrak{gl}}(2,{\mathbb r } )    \end{aligned}\ ] ]    if a subalgebra of @xmath186 is nonsolvable , then it is either three - dimensional and coincides with @xmath188 , or four - dimensional and is equal to the whole of @xmath186 . any solvable irreducible subalgebra is commutative . if it is one - dimensional , then , as follows from the classification of real jordan normal forms of @xmath189 matrices , it is conjugate to the subalgebra  ( i ) . if @xmath45  is two - dimensional , it coincides with the centralizer of one of the jordan normal forms , which implies that it is conjugate to the subalgebra  ( ii ) . if we identify @xmath186 and @xmath184 , the subalgebras listed in lemma  [ lem2 ] are identified with the following subspaces @xmath190 : @xmath191 { \rm(iii)\ } & \langle x^2 , z^2 , 2y - xz\rangle & { \rm(iv)\ } & \langle x^2 , xz , z^2 , y\rangle\end{aligned}\ ] ] consider separately each one of these cases :    ( i)it is easily verified that in this case we have @xmath192 . therefore @xmath193 , and we arrive at the algebra which is listed in the theorem under the number  @xmath194 . ( ii)here we have @xmath195 and the action of the subalgebra  @xmath102 on this space is irreducible . therefore , the space @xmath196 is either zero or coincides with the whole of  @xmath197 . in the former case we immediately find that @xmath193 ( subalgebra  @xmath198 ) . in the second case the subalgebras @xmath199 and @xmath200 coincide and are equal to the subalgebra  @xmath201 of the theorem . ( iii)here @xmath202 , and the @xmath102module @xmath203 is irreducible . hence either we have @xmath204 and then @xmath193 ( subalgebra  @xmath205 ) , or @xmath206 . in the latter case , however , the space  @xmath207 generates a finite - dimensional subalgebra . ( iv)here @xmath208 , and the @xmath102module @xmath197 is a sum of two irreducible submodules @xmath209 and  @xmath210 of the form @xmath211 the submodule  @xmath209 generates a finite - dimensional subalgebra , so that either @xmath204 or @xmath212 . in the former case @xmath193 ( subalgebra  @xmath213 ) , while in the latter the subalgebras @xmath199 and @xmath200 coincide and are equal to the subalgebra  @xmath214 of the theorem . in order to classify all finite - dimensional irreducible contact lie algebras , it will suffice to describe all filtered lie algebras whose associated graded lie algebras are listed in theorem  [ thm1 ] . to solve this latter problem , we shall need the following result . [ pas : l3 ] let @xmath45 be a finite - dimensional filtered lie algebra , and @xmath100 the associated graded lie algebra . if there is an element @xmath215 such that @xmath216=px_p\quad\forall x_p\in{\mathfrak h}_p\ ] ] then @xmath100 , viewed as a filtered lie algebra , is isomorphic to  @xmath45 . suppose @xmath217 for some @xmath218 . for every @xmath130 , consider the subspace @xmath219=px\}.\ ] ] it is easy to show that , @xmath220 for all @xmath130 . thus , the subspace @xmath221 may be identified with  @xmath222 , and since @xmath223\subset{\mathfrak g}^{i+j}(\bar e)$ ] , this identification is in agreement with the structure of the lie algebras @xmath45 and  @xmath100 . hence , we have found an isomorphism of the lie algebras @xmath45 and  @xmath100 which is compatible with their filtrations .    for the graded lie algebras listed in theorem  [ thm1 ] under the numbers @xmath214 , @xmath213 , @xmath201 ,  @xmath198 , we can choose  @xmath124 to be equal to @xmath224 , as this element is contained in all of these algebras . then , in view of  lemma  [ pas : l3 ] , the description of the corresponding filtered lie algebras in these four cases is trivial . consider the remaining two cases @xmath205 and  @xmath194 . @xmath205let @xmath100 be the graded lie algebra that appears under the number  @xmath205 in theorem  [ thm1 ] , and let @xmath45 be a contact lie algebra whose associated graded lie algebra is isomorphic to  @xmath100 . since @xmath225 , the subalgebra  @xmath65 can be identified with the subalgebra  @xmath102 , which is isomorphic to @xmath188 . consider the @xmath65module  @xmath45 . it is completely reducible , and its decomposition into a sum of irreducible submodules has the form : @xmath226 , where the submodule  @xmath227 is one - dimensional and is a complement of  @xmath84 , while the submodule  @xmath228 is two - dimensional and complements  @xmath65 in  @xmath84 . therefore , the submodules @xmath229 , @xmath230 can be identified with the subspaces  @xmath231 of the graded lie algebra  @xmath100 , which allows to identify @xmath45 and  @xmath100 as vector spaces . the structure of the lie algebra  @xmath45 is completely determined by the mappings @xmath232 and @xmath233 defined as restrictions of the bracket operation in  @xmath45 to the corresponding subspaces . the jacobi identity shows that these mappings are both @xmath65invariant . since the @xmath65module @xmath234 is one - dimensional and trivial , we have @xmath235 . similarly , @xmath236 . now , computing the jacobi identity for the basis vectors of @xmath227 and  @xmath228 , we find that the mapping  @xmath237 is zero . thus , the identification of the spaces @xmath45 and  @xmath100 is in agreement with the lie algebra structures of these spaces , so that the lie algebra  @xmath45 is isomorphic to  @xmath100 , viewed as a filtered lie algebra . @xmath194as in the above case , we can identify @xmath65 and  @xmath102 . now , since the @xmath102modules @xmath102 and @xmath103 are not isomorphic for any value of  @xmath237 , we conclude that @xmath84 contains a @xmath65invariant subspace  @xmath228 which is a complement of  @xmath65 . choose a basis  @xmath238 for  @xmath65 and a basis @xmath239 for  @xmath228 in such a way that @xmath240&=\alpha u_1-u_2\\    [ e , u_2]&=u_1+\alpha u_2.\end{aligned}\ ] ] then the elements @xmath241 , together with the element @xmath242 $ ] , will , obviously , form a basis of  @xmath45 , and @xmath243=2\alpha u_3 $ ] . furthermore , checking the jacobi identity , we find that in case @xmath244 we have @xmath245=[u_1,u_3]=0 $ ] , and the lie algebra  @xmath45 is isomorphic to  @xmath100 , viewed as a filtered lie algebra . if @xmath246 , we have @xmath245= \beta u_3 $ ] and @xmath247=-\beta u_2 $ ] for some @xmath248 . note that the parameters @xmath249 and  @xmath250 with @xmath251 give here isomorphic lie algebras , whatever the value of  @xmath252 may be . therefore , up to isomorphism of contact lie algebras we may assume that @xmath253 . if @xmath254 , we find that @xmath45 is again isomorphic to  @xmath100 , viewed as a filtered lie algebra . if @xmath255 or @xmath256 , the lie algebra  @xmath45 is isomorphic to @xmath186 or  @xmath257 respectively , while the subalgebras  @xmath45 can be written , under this identification , in matrix form as follows : @xmath258    summing up what has been said , we obtain the following result :    any finite - dimensional irreducible contact lie algebra is isomorphic to one and only one of the following :    * any of the graded contact lie algebras listed in theorem  [ thm1 ] , if they are viewed as filtered lie algebras ; * @xmath259 , where @xmath260 for @xmath109 , @xmath261 * @xmath262 , where @xmath260 for @xmath109 , @xmath263    from now on , to refer to irreducible contact algebras of type  i , we shall employ the notation i.@xmath264 , where @xmath264 is the number of the corresponding graded contact lie algebra in theorem  [ thm1 ] . [ [ section ] ]    now we shall find explicit representations in contact vector fields for the lie algebras of vector fields described above . note that the mapping @xmath28 that maps an arbitrary function  @xmath265 of the variables @xmath136 into the vector field whose characteristic function is  @xmath265 , defines an embedding of the lie algebra @xmath135 into the algebra of all contact vector fields . in this way we immediately obtain the explicit representations in vector fields for those contact algebras  @xmath45 which are isomorphic to their corresponding graded algebras . in particular , from the representations ( a ) and  ( b ) it follows that these two algebras of contact vector fields can both be embedded into the 10dimensional algebra  i.1 and into the 8dimensional algebra  i.4 . the representations  ( c ) are notable for the fact that the characteristic functions here are independent of  @xmath272 . consider the set of all contact vector fields of the form  @xmath26 , where the function  @xmath265 has the form @xmath273 with @xmath274 and @xmath275  being an arbitrary function of @xmath276 . it is easy to show that this condition is equivalent to the requirement that @xmath26  be an infinitesimal symmetry of the one - dimensional distribution  @xmath47 generated by the vector field @xmath277 . thus we see that this space of vector fields forms an infinite - dimensional subalgebra  @xmath278 of the lie algebra of all contact vector fields . consider the projection @xmath279 given by @xmath280 it is easily verified that this mapping is a homomorphism of lie algebras whose kernel is one - dimensional and is generated by  @xmath281 , while its image coincides with the set of all vector fields on the plane that preserve , up to a constant factor , the volume form @xmath282 on the plane : @xmath283    thus , with every lie algebra of contact vector fields that preserves a one - dimensional distribution complementary to the contact one , we can associate a lie algebra of vector fields on the plane . conversely , the inverse image of any subalgebra of the lie algebra  ( [ eq : pr ] ) of vector fields on the plane is some lie algebra of contact vector fields in the jet space . note that all irreducible lie algebras of contact vector fields , except i.1 and  i.4 , preserve a one - dimensional distribution complementary to the contact distribution , and hence can be embedded into  @xmath278 . the corresponding lie algebras of vector fields on the plane are as follows :    * the lie algebra corresponding to the group of affine transformations of the plane ; * the lie algebra corresponding to the group of equi - affine transformations of the plane ( ie , area - preserving affine transformations ) ; * the lie algebra corresponding to the group of similitude transformations ; * @xmath284 , @xmath285 ( if @xmath246 , this lie algebra corresponds to the group of euclidean transformations ) ; * the lie algebra corresponding to the group of all transformations of the hyperbolic plane ; * the lie algebra corresponding to the group of rotations of the sphere . the above correspondence allows to describe without any difficulty all differential and integral invariants for all lie algebras  @xmath45 of contact vector fields that satisfy the following conditions :      indeed , let @xmath287 be the standard coordinate system in the space @xmath288 of @xmath264th jets of curves on the plane . ( see  @xcite for definition of jet spaces and notions of differential and integral invariants . ) denote by @xmath289 the @xmath264th prolongation of the lie algebra @xmath45 . it then follows from the condition  ( b ) that the differential and integral invariants of  @xmath45 that have the order  @xmath264 are independent of  @xmath272 and may be considered on the manifold of the trajectories of the vector field  @xmath290 . these trajectories are given by the equations @xmath291 and can be parametrized by the coordinates @xmath292 . furthermore , it turns out that if @xmath293 , the action of the algebra  @xmath289 on that quotient manifold is equivalent to the action of the lie algebra @xmath294 on the space of @xmath295th jets , and the mapping @xmath296 that establishes this equivalence has the form : @xmath297 therefore , all differential and integral invariants of  @xmath45 may be derived from the invariants of  @xmath298 by substituting @xmath299 instead of  @xmath300 for @xmath301 . we remark that sophus lie  @xcite found all invariants for those lie algebras of vector fields on the plane that correspond to the cases i.2 and  i.3 . the invariants of the 10dimensional irreducible lie algebra  i.1 were computed in  @xcite over the complex numbers , and they remain unchanged on passing to the real case . now we shall specify nontrivial integral and differential invariants of the least order for the rest of irreducible contact lie algebras of vector fields ; all other invariants can be derived from these by means of differentiation ( see  @xcite ) . @xmath302 \text{i.5 } & \frac{(1+y_2 ^ 2)y_4 - 3y_2y_3 ^ 2}{y_2 ^ 2 } & \frac{y_3dx}{1+y_2 ^ 2 } \\[2 mm ] \text{i.6 } & \frac{y_3e^{-\beta{\operatorname{arctg}}y_2}}{(1+y_2 ^ 2)^{3/2 } } & e^{\beta{\operatorname{arctg}}y_2}(1+y_2)^{1/2}dx \\[2 mm ] \text{ii.1 } & \frac{\sqrt{1+y_1 ^ 2}((1+y_1 ^ 2)y_3 - 3y_1y_2 ^ 2-y_1(1+y_1 ^ 2)^2)}{((1+y_1 ^ 2)^2+y_2 ^ 2)^{3/2 } } & \left(\frac{(1+y_1 ^ 2)^2+y_2 ^ 2}{1+y_1 ^ 2}\right)^{1/2}dx \\[2 mm ] \text{ii.2 } & \frac{\sqrt{1-y_1 ^ 2}((1-y_1 ^ 2)y_3 + 3y_1y_2 ^ 2+y_1(1-y_1 ^ 2)^2)}{((1-y_1 ^ 2)^2+y_2 ^ 2)^{3/2 } } & \left(\frac{(1-y_1 ^ 2)^2+y_2 ^ 2}{1-y_1 ^ 2}\right)^{1/2}dx\\[2 mm ] \hline \end{array}\ ] ] where @xmath303 for the algebras ii.1 and  ii.2 , we have chosen here their representations in contact vector fields that appear earlier under the letter  ( c ) . notice that all contact lie algebras listed in the table above are reducible over the field of complex numbers . hence , for each of these algebras there exists a certain complex analytic contact transformation which takes it to one of the known canonical forms for contact lie algebras over @xmath304 . thus , the inverse thatsformation ( prolonged as many times as needed ) brings known invariants to the invariants of the initial lie algebra . for example , the contact transformation @xmath305 takes the contact lie algebra i.5 to the algebra with the following space ( over @xmath304 ) of characteristic functions : @xmath306 this contact lie algebra is reducible and is the first prolongation of the following lie algebra of vector fields on the plain : @xmath307 the differential invariants of the least order for this lie algebra were computed already ( see , for example ,  @xcite ) and have the form :       the third prolongation of the inverse transformation @xmath308 takes these invariants to those given in the table above . in the similar way we can compute invariants for other contact lie algebras of vector fields given in the table . consider the problem of classifying those infinite - dimensional subalgebras in the lie algebra of contact vector fields that correspond to lie pseudo - groups of contact transformations ( ie , those that can be defined with the help of a finite number of differential equations ; see  @xcite ) . as in the finite - dimensional case , all these algebras can be naturally divided into two classes : reducible ones , which are actually extensions of infinite - dimensional lie algebras of vector fields on the plane , and irreducible ones . over the field of complex numbers all irreducible infinite - dimensional lie algebras of contact vector fields were described by sophus lie  @xcite , who showed that , apart from the lie algebra of all contact vector fields , there exist exactly two infinite - dimensional irreducible subalgebras , namely , the lie algebra @xmath309 , which we already mentioned earlier , and its commutant @xmath310=\{x_{g(x , z)}\}$ ] . the methods for the description of contact lie algebras that have been developed in this paper , can be easily generalized to the infinite - dimensional case . in particular , with these lie algebras we can again associate graded contact lie algebras that can be embedded into the universal extension  @xmath135 of the three - dimensional heisenberg algebra . let @xmath100 be an infinite - dimensional graded subalgebra in  @xmath135 such that @xmath311 for @xmath118 and such that the @xmath102module @xmath103 is irreducible . then all possible types of subalgebras  @xmath102 over the real numbers are listed in lemma  [ lem2 ] . as follows from the proof of theorem  [ thm1 ] , in the cases ( i ) and ( ii ) the lie algebra  @xmath100 is finite - dimensional . the consideration of the remaining cases ( iii ) and  ( iv ) is the same over the complex and real numbers , and gives the infinite - dimensional lie algebras of contact vector fields described above . thus the classification of irreducible infinite - dimensional lie algebras of contact vector fields remains unchanged on passing from the complex to the real case . , _ klassifikation und integration von gewhnlichen differentialgleichungen zwischen @xmath312 , die eine gruppe von transformationen gestatten _ , i vi , from : `` gesamelte abhandlungen '' , volume 5 , teubner , leipzig ( 1924 ) 240310 , 362427 , 432448
the paper is devoted to the complete classification of all real lie algebras of contact vector fields on the first jet space of one - dimensional submanifolds in the plane . 
 this completes sophus lie s classification of all possible lie algebras of contact symmetries for ordinary differential equations . as a main tool 
 we use the abstract theory of filtered and graded lie algebras . 
 we also describe all differential and integral invariants of new lie algebras found in the paper and discuss the infinite - dimensional case .