Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta

Normal forms for pseudo-Riemannian 2-dimensional metrics whose geo desic flo ws admit in tegrals quadratic in m o men ta Alexey V. Bolsino v ∗ , Vladimir S. Matv eev † , Giusepp e Pucac co ‡ Abstract W e discuss pseudo-Riemannian metrics on 2-dimensional manifolds su ch that the geod e sic flow admits a nontrivial integral quadratic in velocities. W e construct (Theorem 1) lo ca l normal forms of such metrics. W e show th at these metrics h a ve certain useful prop erties similar to those of Riemannian Liouville metrics, namely: • they admit geod esically equ iv alen t metrics (Theorem 2); • one can use them to construct a large family of natural systems admitting integ rals quadratic in momenta (Theorem 4); • the integrabil ity of such systems can b e generalized to the qu antum setting (Theorem 5); • these natural systems are integrable by q uadratures (Section 2.2.2). 1 In tro duction Consider a pseudo-Riemannian metric g = ( g ij ) on a s ur face M 2 . A function F : T ∗ M → R is called an in tegral of the geodes ic flo w of g , if { H , F } = 0, where H := 1 2 g ij p i p j : T ∗ M → R is the kinetic e nergy corres p o nding to the metr ic . Geometrically , this condition mea ns that the function is cons ta nt on the orbits of the Hamiltonian sys tem with the Hamiltonian H . W e say the in tegral F is quadratic in momenta if, in every lo cal co ordinate system ( x, y ) on M 2 , it has the for m a ( x, y ) p 2 x + b ( x, y ) p x p y + c ( x, y ) p 2 y , (1) with ( x, y, p x , p y ) ca no nical co o rdinates o n T ∗ M 2 . Geometrically , formula (1) means that the restr iction of the integral to every cota ngent spac e T ∗ p M 2 ≡ R 2 is a homogeneo us qua dr atic function. Of course, H itself is an integral quadratic in momenta for g . W e will say that the integral F is nontrivial , if F 6 = const · H for all co nst ∈ R . The main result of this pa p er is Theor e m 1 below, which gives us a list of lo cal normal forms of metrics of signature (+ , − ) whose geo desic flows admit a no ntrivial int egra l quadratic in momenta. F or the Riemannian case (and, therefore, for the sig nature ( − , − )) such metr ic s a re the well-known Liouville metrics. Theorem 1. Supp ose the met r ic g of signatur e (+ , − ) on M 2 admits a nontrivial inte gr al quadr atic in momenta. Then, in a neighb ourho o d of almost every p oint ther e ex ist c o or dinates x, y such that the metr ic and the inte gr al ar e as in the fol lowing table: Liouvil le c ase Complex-Liouvil le c ase Jor dan-blo ck c ase g ( X ( x ) − Y ( y ))( dx 2 − dy 2 ) ℑ ( h ) dxdy (1 + xY ′ ( y )) dxdy F X ( x ) p 2 y − Y ( y ) p 2 x X ( x ) − Y ( y ) p 2 x − p 2 y + 2 ℜ ( h ) ℑ ( h ) p x p y p 2 x − 2 Y ( y ) 1+ xY ′ ( y ) p x p y wher e ℜ ( h ) and ℑ ( h ) ar e t he r e al and imaginary p arts of a holomorph ic function h of the variable z := x + i · y . ∗ Departmen t of Mathematical Sciences, Lough borough Universit y , LE11 3TU UK, A.Bols inov @lb oro.ac.uk † Institute of M athematics, FSU Jena, 07737 Jena Germany , matv eev@minet.uni-jena.de ‡ Dipartiment o di Fisica, Universit` a di Roma “T or V ergata”, 00133 Rome Italy , pucacco@roma2.infn.it 1 Given a metric and the quadratic integral, it is easy to understand wha t case they b elong to . Indeed, for the in tegral (1) the matrix F ij =  a b 2 b 2 c  can be viewed as a (2 , 0)-tensor: if we change the co ordinate sys tem a nd rewr ite the function F in the new co ordinates, the matrix c hanges according to the tensor r ule. Then, G i j := X α g j α F iα (2) is a (1 , 1)-tensor. By dire ct calculatio n we see that G i j has tw o different r eal eig e n v alues in the first ca se, t wo complex-conjugate eigenv alues in the second ca se and is (conjuga te to) a Jordan-blo ck in the third cas e. This also explains our choice of the names for the normal forms of the metrics. Indeed, in the Riemannia n case, the tensor (2) alwa ys has tw o real eigenv alues . In par ticular, the no rmal for m of the Riemannian metric admitting an integral quadra tic in momenta, which is tra ditio nally called Lio uville form (or Liouville metric), is very s imilar to the metric o f our “Lio uville” case. One can view our “Complex-Lio uville” case as the complexification of the sta nda rd Liouville metric: if in the expr ession ( X ( x ) − Y ( y ))( dx 2 + dy 2 ) we replace X by (a holomor phic function) h ( z ), Y b y h ( z ), dx by dz , a nd dy by id ¯ z , we obtain the Complex- Liouville metric up to the factor 8 i . T he Jordan- blo ck ca se has no dir e ct a nalog in the Riemannian setting. R emark 1 . The co rresp onding na tural Hamiltonia n pr o blem on the hyperb olic pla ne ha s rece ntly b een treated in [39] following an approa ch used by Rosq uist a nd Uggla [40]. Systems with indefinite signature hav e b een investigated b efore in the classical w orks by K alnins and Miller on separa tion of v ar iables [2 0, 38], see also [13, 16, 36]. O ther pos sible approa ches ar e based on Killing tensor theory [5], r -matrix theory [24] and a lgebraic metho ds [15]. F or the cor resp onding quantum ca se w e refer to [19] a nd references therein. R emark 2 . A part, if not all credits for the results of the present pap er should b e given to Dar b o ux, see [14, §§ 5 92–5 9 4,600 –608]. There is no doubt that Darb oux was very close to Theo r em 1, to the results of Section 2.2.2, a nd, to a certa in extent, to Theore m 2 of our pap er, and could g et it if he would hav e b een int erested in the pseudo-Riemannain metrics. More precisely , • In [14, § 5 93], Darb oux gets the Riemannian Lio uville metrics. Since he worked ov er complex co o rdi- nates, his for m ulas can be interpreted as our L iouville and Complex-Liouville cases. • In [14, § 594 ], Darb oux gets (a case that could be interpreted as) the J o rdan-blo ck case. • T he formulas o f Sectio n 2 .2.2 o f the present pap er are simila r to that of [14, § 594 ]. How ev er, Darb oux was interested in the po s itive definite metrics only . Actually , in his time it was unusual to cons ider indefinite metrics, since the applica tions of pseudo-Riema nnia n metrics to general re lativity and cosmolog y app eared muc h later . Dar bo ux worked ov er complex co o r dinates x, y and explicitly rema r ked on the tr ansformatio n x = u + iv , y = u − iv leading to the standard metric of the (+,+) cas e, with no mention of a p o s sible interpretation of x, y as real co or dinates. The o nly e x ception is the Jo r dan-blo ck case with constant function Y (equations (24,2 5) o f [1 4, § 5 9 4]), where o ne c an get the surfa c es of revolution. The r esults of this paper were announced in [10]. 2 Applications 2.1 Applications in geometry: normal forms for 2-dimensional geo desically equiv alen t metr ics Two metrics g and ¯ g on o ne manifold are geo des ically equiv alent, if e very (unparametr ized) g eo desic of the firs t metric is a g eo desic o f the seco nd metric. In vestigation of g eo desically eq uiv alent metrics is a 2 classical pr oblem in differential geometry , see the sur veys [3, 33, 37] or/a nd the introductions to [3 1, 32, 3 4]. In particular , normal forms for g eo desically equiv a lent Riemannian 2 -dimensional metrics were alre a dy constructed by Dini [17]. An e a sy cor o llary of Theor e m 1 is the following theorem which gives norma l forms of geo desically equiv a lent nonprop ortiona l metrics such that one of them has signature (+ , − ). Theorem 2. L et g , ¯ g b e ge o desic al ly e quivalent metrics on M 2 such that g has signatur e (+ , − ) , and ¯ g 6 = c onst · g for every c onst ∈ R . Then, in a neighb ourho o d of almost every p oint, ther e exist c o or dinates such t hat metr ics ar e as in the fol lowing table: Liouvil le c ase Complex-Liouvil le c ase Jor dan-blo ck c ase g ( X ( x ) − Y ( y ))( dx 2 − dy 2 ) ℑ ( h ) dxdy (1 + xY ′ ( y )) dxdy ¯ g  1 Y ( y ) − 1 X ( x )   dx 2 X ( x ) − dy 2 Y ( y )  −  ℑ ( h ) ℑ ( h ) 2 + ℜ ( h ) 2  2 dx 2 + 2 ℜ ( h ) ℑ ( h ) ( ℑ ( h ) 2 + ℜ ( h ) 2 ) 2 dxdy +  ℑ ( h ) ℑ ( h ) 2 + ℜ ( h ) 2  2 dy 2 1+ xY ′ ( y ) Y ( y ) 4  − 2 Y ( y ) dxdy + (1 + xY ′ ( y )) dy 2  wher e h is holomorphi c function of the variable z := x + i · y . R emark 3 . It it natural to co nsider the metrics from the Complex-Liouville case as the complexification of the metrics fro m the Liouville case: indeed, in the complex co o rdinates z = x + i · y , ¯ z = x − i · y , the metrics hav e the form ds 2 g = − 1 8 ( h ( z ) − h ( z ))  d ¯ z 2 − dz 2  , ds 2 ¯ g = − 1 4  1 h ( z ) − 1 h ( z )   d ¯ z 2 h ( z ) − dz 2 h ( z )  . R emark 4 . In the Jor dan-blo ck case , if d Y 6 = 0 (which is alwa ys the case at almost every p oint, if the restriction of g to any neighborho o d do es not admit a K illing v ector field), after a lo ca l co or dinate change, the metrics g and ¯ g hav e the form (s e e a lso Remark 14) ds 2 g =  ˜ Y ( y ) + x  dxdy ds 2 ¯ g = − 2( ˜ Y ( y ) + x ) y 3 dxdy + ( ˜ Y ( y ) + x ) 2 y 4 dy 2 . Pro of of Theorem 1 . W e will use the next theor e m which probably was alrea dy k nown to Dar bo ux [14, § 608]. F or recent pro ofs, see [2 6 , 2 7, 28, 44]. Theorem 3. L et g b e a metric on M 2 and h ∈ Γ( S 2 M 2 ) b e a symmetric n onde gener ate biline ar form on M 2 . Consider the fol lowing metric ¯ g =  det( g ) det( h )  2 h (3) on M 2 . If g and ¯ g ar e ge o desic al ly e quivalent, then the function ˆ h : T M → R , ˆ h ( ξ ) := h ( ξ , ξ ) is an inte gr al for t he ge o desic flow of g . R emark 5 . Theor em 3 and Cor ollary 1 be low bea r some resemblance with other classe s of tra nsformations betw een dynamica l s ystems [1, 2, 11, 38, 41, 42, 4 3]. How ever, the present r e sult is of different nature and is dee per b ecause, in o rder to construct the second system, o ne needs to know the quadratic integral o f the first one. Combining Theor em 3 with Theor em 1, we obtain tha t, in a neighbourho o d of almost every p oint, geo desically equiv alent metrics g and ¯ g are as in the table in Theo rem 2 (we assume that g has s ig nature (+ , − ) a nd that ¯ g 6 = co ns t · g ). Thus, in or der to prov e Theorem 2, we need to show that the metrics from the table are indeed g eo desically equiv alen t, which can b e done by direct calcula tions. Indeed, it is well-kno wn, 3 see for example [18, § 40 of Ch. I I I], that t wo metrics a re g eo desically equiv alen t if a nd only if the difference of their Levi-C ` ıvita connec tions has the form Υ j δ i k + Υ k δ i j for a one-for m Υ = (Υ i ). Direct calcula tio n o f the Levi-C ` ıvita connections for the metrics shows that it is indeed the case : the form Υ equals 1 2  X ′ ( x ) X ( x ) dx + Y ′ ( y ) Y ( y ) dy  for the normal forms of the metrics in the Liouville case, ℑ ( h ) ∂ ∂ x ℑ ( h ) + ℜ ( h ) ∂ ∂ y ℑ ( h ) ( ℑ ( h )) 2 + ( ℜ ( h )) 2 dx + ℑ ( h ) ∂ ∂ y ℑ ( h ) − ℜ ( h ) ∂ ∂ x ℑ ( h ) ( ℑ ( h )) 2 + ( ℜ ( h )) 2 dy for the complex Lio uville case and Y ′ ( y ) Y ( y ) dy fo r the Jorda n- blo ck case. Corollary 1. Le t g b e a m et ric on M 2 and h ∈ Γ( S 2 M 2 ) b e a symmetric nonde gener ate biline ar form on M 2 . Then, g and the metric (3) ar e ge o desic al ly e quivalent, if and only if the function ˆ h : T M → R , ˆ h ( ξ ) = h ( ξ , ξ ) is an inte gr al for t he ge o desic flow of g . Pro of. In the directio n “= ⇒ ” the statement coincides with Theorem 3 . In order to prove in “ ⇐ =” direction, it is sufficient to chec k the statement in the neighbourho o d of a lmo st every point. Here, the metrics g , ¯ g a nd the integrals ˆ h are g iven by Theor ems 1,2 and are related precisely by formula (3). R emark 6 . Theor em 3 had found a recent imp ortant application in the solution of tw o problems explicitly stated by Sophus Lie in [25] due to [12 , 3 5]. 2.2 Applications in mathematical physics 2.2.1 Natural system s admi tting an integra l quadratic in m oment a F or a pseudo- Riemannian manifold ( M , g ), a natural Hamil tonian syste m is a Hamitonian system with H : T ∗ M → R o f the form H := H g + U = 1 2 g ij p i p j + U ( x, y ). W e s ay that a natural Hamilto nia n system is quadratically i n tegrable , if there exists a function F of the form F = F g + V = F ij p i p j + V ( x, y ) suc h that { H , F } = 0 with F 6 = const 1 · H + co nst 2 for all co nst 1 , const 2 ∈ R . R emark 7 . In [3 9], the natural Ha miltonian system on the Minko wski plane has b een reduced to the corre- sp onding kinetic Hamiltonian system with conformal (J a cobi) pseudo-Euclidean metric. Theorem 4. L et g b e a metric of signatur e (+ , − ) on M 2 . Assume a natu r al Hamiltonian s yst em with Hamiltonian H g + U to b e qu adr atic al ly inte gr able with inte gr al F = F g + V . Then, in a neighb ourho o d of almost every p oint, ther e exists a c o or dinate system such that the metric g and t he funct ions F g , U , V ar e as in the fol lowing table: Liouvil le c ase Complex-Liouvil le c ase Jor dan-blo ck c ase g ( X ( x ) − Y ( y ))( dx 2 − dy 2 ) ℑ ( h ) dxdy (1 + xY ′ ( y )) dxdy F g X ( x ) p 2 y − Y ( y ) p 2 x X ( x ) − Y ( y ) p 2 x − p 2 y + 2 ℜ ( h ) ℑ ( h ) p x p y , p 2 x − 2 Y ( y ) 1+ xY ′ ( y ) p x p y U 1 2 ˆ X ( x ) − ˆ Y ( y ) X ( x ) − Y ( y ) ℑ ( h 1 ) ℑ ( h ) xY ′ 1 ( y )+ Y 2 ( y ) 1+ xY ′ ( y ) V ˆ Y ( y ) X ( x ) − ˆ X ( x ) Y ( y ) X ( x ) − Y ( y ) ℜ ( h ) ℑ ( h 1 ) ℑ ( h ) − ℜ ( h 1 ) − Y xY ′ 1 ( y )+ Y 2 ( y ) 1+ xY ′ ( y ) + Y 1 ( y ) wher e h, h 1 ar e holomorphic functions of the variable z := x + i · y . Pro of. It is well known (s e e, for example, [6]), that the condition { H , F } = 0 is in this case equiv alent to the following t wo conditions: { H g , F g } = 0 , (4) 2 dU ◦ G = dV , (5) 4 where G is given by (2). In tensor index notations, (5) is 2 G i j ∂ U ∂ x i = ∂ V ∂ x j . (6) Indeed, condition { H , F } = 0 is equiv a lent to the following eq uation: { H g , F g } + { H g , V } − { F g , U } = 0 . Since { H g , F g } (resp ectively , { H g , V } − { F g , U } ) is a thir d degree-p oly nomial in mo menta (respe ctively , first degree), the latter equation is equiv alent to: { H g , F g } = 0 (7) { F g , U } = { H g , V } . (8) W e see that (7) coincides with (4) and (8) is equiv alent to 2 F ij ∂ U ∂ x i = g ij ∂ V ∂ x i , which is equiv a lent to (6) and therefore to (5). Condition (4) tells us that the function F g is an integral quadratic in momenta for the geo desic flow of g . Clearly , F g is nontrivial. Indee d, if F g = const 1 · H g , then condition (5) rea ds co nst 1 ◦ dU = dV implying V = const 1 · U + const 2 . These in turn imply F = const 1 · H + const 2 , which contradicts the assumptions. Thu s, F g is a non trivial in tegra l of the geo desic flow of the metric g . By Theo rem 1, almost every p oint has a neighbour ho o d with lo cal co ordinates ( x, y ) such that g a nd F g are a s in the table. In or der to prov e Theorem 4 , it is sufficient to show that, for every co lumn of the table, the functions U and V are c omplete solutions of e q uation (5). Her e we consider the three cases in detail. Liouville case. Assume g , F g are a s in the fir st co lumn of the table. Then the form dU ◦ G is − Y ( y ) ∂ U ∂ x dx − X ( x ) ∂ U ∂ y dy and c ondition (5) reads ( ∂ Y ( y ) U ∂ x = − 1 2 ∂ V ∂ x , ∂ X ( x ) U ∂ y = − 1 2 ∂ V ∂ y . (9) Different iating the second equa tion w.r.t. x and subtra cting the der iv a tive of the fir st equation w.r .t. y , we obtain 0 = ∂ ∂ x  X ( x ) ∂ U ∂ y  − ∂ ∂ y  Y ( y ) ∂ U ∂ x  = ∂ 2 ( X ( x ) − Y ( y )) U ∂ x∂ y implying U = 1 2 ˆ X ( x ) − ˆ Y ( y ) X ( x ) − Y ( y ) for certain functions ˆ X = ˆ X ( x ) a nd ˆ Y = ˆ Y ( y ). Subs tituting U in (9), we obtain V = X ( x ) ˆ Y ( y ) − Y ( y ) ˆ X ( x ) X ( x ) − Y ( y ) . Thu s, in the Liouville case, U and V are as in the table. Complex-Liouvi lle case. In this case 2 dU ◦ G is equal to  ℜ ( h ) ∂ U ∂ x − ℑ ( h ) ∂ U ∂ y  dx +  ℑ ( h ) ∂ U ∂ x + ℜ ( h ) ∂ U ∂ y  dy =  ∂ ℜ ( h ) U ∂ x − ∂ ℑ ( h ) U ∂ y  dx +  ∂ ℜ ( h ) U ∂ y + ∂ ℑ ( h ) U ∂ x  dy 5 and c ondition (5) is equiv alent to the following sys tem o f PDE : ( ∂ ℜ ( h ) U ∂ x − ∂ ℑ ( h ) U ∂ y = ∂ V ∂ x , ∂ ℜ ( h ) U ∂ y + ∂ ℑ ( h ) U ∂ x = ∂ V ∂ y . (10) W e see that these eq ua tion are pr ecisely the Cauch y-Riemann condition for the function h 1 := ℜ ( h ) U − V + i · ℑ ( h ) U . Thus, U = ℑ ( h 1 ) ℑ ( h ) and V = ℜ ( h ) U − ℜ ( h 1 ) = ℜ ( h ) ℑ ( h 1 ) ℑ ( h ) − ℜ ( h 1 ) . W e see that U and V ar e as in the table. Jordan-blo c k case. In this case the 1-form 2 dU ◦ G is − Y ( y ) ∂ U ∂ x dx +  (1 + xY ′ ( y )) ∂ U ∂ x − Y ( y ) ∂ U ∂ y  dy and c ondition (5) is equiv alent to the following sys tem o f PDE :  − Y ( y ) ∂ U ∂ x = ∂ V ∂ x , (1 + xY ′ ( y )) ∂ U ∂ x − Y ( y ) ∂ U ∂ y = ∂ V ∂ y . (11) The first equation in (11) is equiv alen t to V = − Y ( y ) U + Y 1 ( y ). Substituting this in the seco nd eq uation, we obtain (1 + xY ′ ( y )) ∂ U ∂ x − Y ( y ) ∂ U ∂ y = − ∂ Y ( y ) U ∂ y + Y ′ 1 ( y ) which implies ∂ (1 + xY ′ ( y )) U ∂ x = Y ′ 1 ( y ) and ther efore (1 + xY ′ ( y )) U = xY ′ 1 ( y ) + Y 2 ( y ). Thus, U = xY ′ 1 ( y ) + Y 2 ( y ) 1 + xY ′ ( y ) and V = − Y xY ′ 1 ( y ) + Y 2 ( y ) 1 + xY ′ ( y ) + Y 1 ( y ) . 2.2.2 In tegration b y quadratures of natural systems admitting an integral qua dratic in momenta Since the time of Ja cobi it is known that (in the 2-dimensio nal Riemannian c a se) nontrivial integrals quadratic in momenta ar e ex tremely helpful for the des cription of dyna mics o f natural systems: indeed, in this case • the Hamilton equations , which are a system of four ODE on T ∗ M 2 , ca n b e reduced to a para meter- depe nding system of tw o ODE on M 2 . • Mo reov er, it is po ssible to construct a c haracteristic (= function constant on the solutions) o f this system b y means of the integration of certain functions of one v ariable only . 6 See [7, 45] for deta ils. Classically , the second prop erty is referr ed to as “the s ystem is int egrable by quadratures”. Both prop erties are use ful for exact solutions, fo r numerical analysis a nd for a qualitative description of (the solutions of ) the Hamilton equations. W e a re going to show that these nice prop erties p ers ist in the pseudo- Riemannian setting. Liouville case. There is virtually no difference with resp ect to the Riemannian s etting. Consider H = H g + U and F = F g + V such that g , F g , U, V are as in the first column of the table fr o m Theo rem 4. Then, the first tw o Hamilton equations are ( d dt x = ∂ H ∂ p x = p x X − Y , d dt y = ∂ H ∂ p y = − p y X − Y . (12) Since the functions F and H are consta n t on the solutions of the system, for every p oint ( x, y , p x , p y ) of the solution w e hav e    1 2 p 2 x − p 2 y X ( x ) − Y ( y ) + 1 2 ˆ X ( x ) − ˆ Y ( y ) X ( x ) − Y ( y ) = H 0 , X ( x ) p 2 y − Y ( y ) p 2 x X ( x ) − Y ( y ) + ˆ Y ( y ) X ( x ) − ˆ X ( x ) Y ( y ) X ( x ) − Y ( y ) = F 0 . This is a linea r sys tem o n p 2 x , p 2 y , solving it w.r .t. p x and p y we obtain  p 2 x = 2 H 0 X ( x ) + F 0 − ˆ X ( x ) , p 2 y = 2 H 0 Y ( y ) + F 0 − ˆ Y ( y ) . (13) Substituting these in (12), we obtain    d dt x = ε 1 √ 2 H 0 X ( x )+ F 0 − ˆ X ( x ) X − Y := v 1 , d dt y = ε 2 √ 2 H 0 Y ( y )+ F 0 − ˆ Y ( y ) X − Y := v 2 . (14) W e see that Hamilton equations can b e reduced to a system of tw o ODE on M 2 depe nding on the pa rameters H 0 , F 0 ∈ R and ε i ∈ {− 1 , +1 } . Clearly , a function K ( x, y ) is a characteristic of the sys tem (14) if dK v a nishes on the vector field v := ( v 1 , v 2 ). Since the fo rm B := ε 1 dx q 2 H 0 X ( x ) + F 0 − ˆ X ( x ) − ε 2 dy q 2 H 0 Y ( y ) + F 0 − ˆ Y ( y ) v anis hes o n v and is closed, the function K ( p ) := Z p p 0 B = Z x x 0 dξ q 2 H 0 X ( ξ ) + F 0 − ˆ X ( ξ ) − ε 1 ε 2 Z y y 0 dξ q 2 H 0 Y ( ξ ) + F 0 − ˆ Y ( ξ ) is a characteristic. W e s e e that in o rder to find a characteristic, we only need to integrate tw o functions of one v aria ble e a ch, i.e., the sy stem is integrable by quadratur e s. Complex-Liouvi lle case. Cons ider H = H g + U a nd F = F g + V such that g , F g , U, V are as in the second column of the table fr om The o rem 4. Then, the firs t tw o Hamilto n equations are ( d dt x = ∂ H ∂ p x = 2 p y ℑ ( h ) , d dt y = ∂ H ∂ p y = 2 p x ℑ ( h ) . (15) Since the functions F and H are consta n t on the solutions of the system, for every p oint ( x, y , p x , p y ) of the solution w e hav e ( 2 p x p y ℑ ( h ) + ℑ ( h 1 ) ℑ ( h ) = H 0 , p 2 x − p 2 y + ℜ ( h )  2 p x p y ℑ ( h ) + ℑ ( h 1 ) ℑ ( h )  − ℜ ( h 1 ) = F 0 . 7 Subtracting the first eq uation times ℜ ( h ) fr om the s econd, w e obtain  2 p x p y = H 0 ℑ ( h ) − ℑ ( h 1 ) , p 2 x − p 2 y = − ( ℜ ( h ) H 0 − ℜ ( h 1 )) + F 0 . F ro m these , adding (resp ectively , substracting) to (resp ectively , fro m) the s e cond equation the first equation times i , w e obtain  ( p x − i · p y ) 2 = − ( H 0 ℜ ( h ) − ℜ ( h 1 ) − F 0 ) − i · ( H 0 ℑ ( h ) − ℑ ( h 1 )) = − H 0 h + h 1 + F 0 , ( p x + i · p y ) 2 = − ( H 0 ℜ ( h ) − ℜ ( h 1 ) − F 0 ) + i · ( H 0 ℑ ( h ) − ℑ ( h 1 )) = − H 0 ¯ h + ¯ h 1 + F 0 . R emark 8 . Since 1 2 ( p x − i · p y ) is the ca no nical momentum conjuga te to z = x + i · y , these equations ar e the complex analog o f (13) . Then, p x = ε ℜ  √ − H 0 h + h 1 + F 0  and p y = − ε ℑ  √ − H 0 h + h 1 + F 0  (the choice of the bra nch of the square ro ot is hidden in ε ). Substituting these in (15), we obtain    d dt x = − 2 ε ℑ ( √ − H 0 h + h 1 + F 0 ) ℑ ( h ) := v 1 , d dt y = 2 ε ℜ ( √ − H 0 h + h 1 + F 0 ) ℑ ( h ) := v 2 . (16) W e see that Hamilton equations can b e reduced to a system of tw o ODE on M 2 depe nding on the pa rameters H 0 , F 0 ∈ R , and ε ∈ {− 1 , +1 } . Consider the 1 -form B := ℜ  √ − H 0 h + h 1 + F 0  | − H 0 h + h 1 + F 0 | dx + ℑ  √ − H 0 h + h 1 + F 0  | − H 0 h + h 1 + F 0 | dy . The Cauch y-Riemann conditions for the ho lomorphic function √ − H 0 h + h 1 + F 0 imply that the fo r m is closed. Clearly , the for m v anishes on the vector field v = ( v 1 , v 2 ). Then, the function K ( p ) := Z p p 0 B = Z x x 0 ℜ  √ − H 0 h + h 1 + F 0  | − H 0 h + h 1 + F 0 | dξ + Z y y 0 ℑ  √ − H 0 h + h 1 + F 0  | − H 0 h + h 1 + F 0 | dξ is co nstant on the solutio ns of (16), i.e., is a characteristic of the system. It is e asy to check by direct calculations that in the co mplex coo rdinate z the form B is 2 ℜ  dz √ − H 0 h + h 1 + F 0  . Thu s, the function K eq uals to 2 ℜ Z z z 0 dξ p − H 0 h ( ξ ) + h 1 ( ξ ) + F 0 ! , i.e., the system is in tegrable b y quadrature s . Jordan-blo c k case. Conside r H = H g + U and F = F g + V such that g , F g , U, V are as in the third column of the ta ble fr om Theorem 4. Then, the first tw o Hamilton eq ua tions are ( d dt x = ∂ H ∂ p x = 2 p y 1+ xY ′ ( y ) , d dt y = ∂ H ∂ p y = 2 p x 1+ xY ′ ( y ) . (17) Since the functions F and H are consta n t on the solutions of the system, for every p oint ( x, y , p x , p y ) of the solution w e hav e    2 p x p y 1+ xY ′ ( y ) + Y 2 ( y )+ xY ′ 1 ( y ) 1+ xY ′ ( y ) = H 0 , p 2 x − Y ( y )  2 p x p y 1+ xY ′ ( y ) + Y 2 ( y )+ xY ′ 1 ( y ) 1+ xY ′ ( y )  + Y 1 ( y ) = F 0 . 8 Adding the firs t equation times Y ( y ) to the second one, we obtain  p 2 x = H 0 Y ( y ) − Y 1 ( y ) + F 0 2 p x p y = x ( H 0 Y ′ ( y ) − Y ′ 1 ( y )) + H 0 − Y 2 ( y ) = ⇒    p x = ε p H 0 Y ( y ) − Y 1 ( y ) + F 0 , p y = ε 2 x ( H 0 Y ′ ( y ) − Y ′ 1 ( y ) ) + H 0 − Y 2 ( y ) √ H 0 Y ( y ) − Y 1 ( y )+ F 0 , where ε ∈ {− 1 , +1 } . Substituting thes e in (17), we obtain      d dt x = ε x ( H 0 Y ′ ( y ) − Y ′ 1 ( y ) ) + H 0 − Y 2 ( y ) (1+ xY ′ ( y )) √ H 0 Y ( y ) − Y 1 ( y )+ F 0 := v 1 , d dt y = ε 2 √ H 0 Y ( y ) − Y 1 ( y )+ F 0 1+ xY ′ ( y ) := v 2 . (18) W e see that Hamilton equations can b e reduced to a system of tw o ODE on M 2 depe nding on the pa rameters H 0 , F 0 ∈ R , and ε ∈ {− 1 , +1 } . Consider the 1 -form B := dx p H 0 Y ( y ) − Y 1 ( y ) + F 0 − 1 2 x ( H 0 Y ′ ( y ) − Y ′ 1 ( y )) − Y 2 ( y ) + H 0 ( H 0 Y ( y ) − Y 1 ( y ) + F 0 ) 3 / 2 dy (19) = d " x p H 0 Y ( y ) − Y 1 ( y ) + F 0 # + 1 2 Y 2 ( y ) − H 0 ( H 0 Y ( y ) − Y 1 ( y ) + F 0 ) 3 / 2 dy . (20) By (20), the form is closed. By (19), the for m v anishes on the v ector field v = ( v 1 , v 2 ). Then, the function K ( p ) := Z p p 0 B = x p F 0 − Y 1 ( y ) + H 0 Y ( y )     p p 0 + 1 2 Z y y 0 Y 2 ( ξ ) − H 0 ( F 0 − Y 1 ( ξ ) + H 0 Y ( ξ )) 3 / 2 dξ is a characteristic of the system (18), i.e . the system is integrable by quadrature s. 2.2.3 Quan tum in tegrability Let g be a metric, and ( F ij ) ∈ Γ( S 2 M 2 ) b e a s y mmetric bilinear 2- fo rm o n T ∗ M 2 . Consider the following t wo linear par tia l differential o p erators ∆ g , F g : C ∞ → C ∞ : ∆ g := − X i,j 1 p | det ( g ) | ∂ ∂ x i g ij p | det ( g ) | ∂ ∂ x j F := X i,j 1 p | det ( g ) | ∂ ∂ x i F ij p | det ( g ) | ∂ ∂ x j R emark 9 . The first op erator is the Beltra mi- Laplace op er ator of the metric g ; a nother wa y to write it down is ∆ g = − X i,j g ij ∇ i ∇ j , where ∇ is the Levi-C ` ıv ita connection of g . The second op er ator is a natural q ua ntization of the function P i,j F ij p i p j and a nother wa y to write it down is F g = X i,j ∇ i F ij ∇ j . In particular, both o p er ators do no t depend o n the choice of the co or dinate sys tem. R emark 10 . The symbols of ∆ g and of F g are − 2 H := − 2 P i,j g ij p i p j and P i,j F ij p i p j , resp ectively . 9 Theorem 5. L et F = P i,j F ij p i p j + V ( x, y ) b e a quadr atic int e gr al of the natur al Hamiltonian system 1 2 P i,j g ij p i p j + U ( x, y ) on T ∗ M 2 . Then, the op er ators H := ∆ g − 2 U and F := F g + V c ommu te: H ◦ F = F ◦ H . R emark 11 . The Riemannian analog of Theo rem 5 fo llows from [9 , 2 3, 29, 30]. Pro of of Theorem 5. It is sufficient to chec k the statement at almost every point, i.e., for the metrics and the integrals from Theorem 4. Direct calculations shows that in this case the o p er ators ∆ g and F g are as in the following table: Liouvil le c ase Complex-Liouvil le c ase Jor dan-blo ck c ase ∆ g − 1 X ( x ) − Y ( y )  ∂ 2 ∂ x 2 − ∂ 2 ∂ y 2  − 4 ℑ ( h ) ∂ 2 ∂ x∂ y − 4 1+ x 1 Y ′ ( y ) ∂ 2 ∂ x∂ y F g 1 X ( x ) − Y ( y )  X ( x ) ∂ 2 ∂ y 2 − Y ( y ) ∂ 2 ∂ x 2  ∂ 2 ∂ x 2 − ∂ 2 ∂ y 2 + 2 ℜ ( h ) ℑ ( h ) ∂ 2 ∂ x∂ y ∂ 2 ∂ x 2 − 2 Y ( y ) 1+ xY ′ ( y ) ∂ 2 ∂ x∂ y where h is a holo mo rphic function of z = x + i · y . T o prov e that H = ∆ g − 2 U and F = F g + V commut e, we first observe that in the Liouville a nd Jordan- blo ck cases: F g + V = ∂ 2 ∂ x 2 + f · (∆ g − 2 U ) + f 1 , where f = X ( x ), f 1 = ˆ X ( x ) fo r the Liouville case , a nd f = Y ( y ) 2 and f 1 = Y 1 ( y ) for the Jordan blo ck case. Similarly , in the co mplex Lio uville case, w e hav e F g + V = ∂ 2 ∂ x 2 − ∂ 2 ∂ y 2 + f · (∆ g − 2 U ) + f 1 where f = − ℜ ( h ) 2 , f 1 = ℜ ( h 1 ) 2 The La place-Beltra mi op e r ator ∆ g in all the cases is of the form ∆ g = λ − 1 ∆ g 0 , where ∆ g 0 is the Laplace-Beltr ami op erator o f the flat metric g 0 (more sp ecifica lly , g 0 is dx 2 − dy 2 in the Liouville case, a nd 2 dxdy in the complex Liouville and Jorda n-blo ck cases ). Using the fact that ∆ g 0 commutes with ∂ ∂ x , it is straightforward to verify the following commutator for m ula: [∆ g − 2 U , ∂ 2 ∂ x 2 ] =  λ xx λ + 2 λ x λ ∂ ∂ x  ◦ (∆ g − 2 U ) + 2 ( λU ) xx λ + 4 ( λU ) x λ ∂ ∂ x (here w e use standard notation for the comm utator of t wo linear op erator s [ A , B ] = A ◦ B − B ◦ A ). The tw o following formulas are standar d: [∆ g − 2 U , f · (∆ g − 2 U )] =  ∆ g f − 2 grad g f  ◦ (∆ g − 2 U ) and [∆ g − 2 U , f 1 ] = ∆ g f 1 − 2 gr ad g f 1 , where the v ector field grad g f is viewed as a first order differential op erato r, i.e., gr ad g f = g ij ∂ f ∂ x i ∂ ∂ x j . Thu s, in the Liouville and J o rdan-blo ck cases , we hav e: [ H , F ] = [∆ g − 2 U , F g + V ] =  λ xx λ + 2 λ x λ ∂ ∂ x + ∆ g f − 2 grad g f  ◦ (∆ g − 2 U )+ + 2 ( λU ) xx λ + 4 ( λU ) x λ ∂ ∂ x + ∆ g f 1 − 2 gra d g f 1 10 Hence, the commut ativity condition [ H , F ] = H ◦ F − H ◦ F = 0 splits into fo ur simple equa tions (here we use the fac t that ∆ g = λ − 1 ∆ g 0 and g rad g = λ − 1 grad g 0 ): (i) λ x ∂ ∂ x − gr ad g 0 f = 0 (ii) λ xx + ∆ g 0 f = 0 (iii) 2( λU ) x ∂ ∂ x − gr ad g 0 f 1 = 0 (iv) 2( λU ) xx + ∆ g 0 f 1 = 0 Each o f thes e equations has na tural meaning. Indeed, (i) and (ii) mea n that the o p er ators ∆ g and F g commute (without p otentials), (iii) and (iv) give the “new” commutativit y conditions inv olving the po ten- tials. The firs t a nd thir d eq uations a re equiv alen t to the commutativit y o f cla ssical int egr a ls, wher e as the second a nd the fourth k eep a dditional “q ua ntu m” informatio n. It is interesting to no tice that the quantum conditions (ii) and (iv) c a n be obtained from the clas sical o nes (i) and (iii) by “differ e ntiating” s o that in our particular case the q uantum int egra bility in dimension 2 tur ns o ut to b e a co rollar y of the clas sical one: λ xx + ∆ g 0 f = div( λ x ∂ ∂ x − gr ad g 0 f ) 2( λU ) xx + ∆ g 0 f 1 = div(2( λU ) x ∂ ∂ x − gr ad g 0 f 1 ) How ev er, each o f the ab ov e four co nditions can b e verified directly . T aking into acco unt the following explicit formulas: Liouville case : ∆ g 0 = − ∂ ∂ x 2 + ∂ ∂ y 2 , grad g 0 f = f x ∂ ∂ x − f y ∂ ∂ y , λ = X ( x ) − Y ( y ) Jordan- blo ck ca se : ∆ g 0 = − 2 ∂ 2 ∂ x∂ y , grad g 0 f = f y ∂ ∂ x + f x ∂ ∂ y , λ = 1 / 2 (1 + xY ′ ( y )) we see that equations(i)–(iv) b ecome: Liouville case: ( λ x − f x ) ∂ ∂ x + f y ∂ ∂ y = 0 λ xx − f xx + f y y = 0 (2( λU ) x − ( f 1 ) x ) ∂ ∂ x + ( f 1 ) y ∂ ∂ y = 0 2( λU ) xx − ( f 1 ) xx + ( f 1 ) y y = 0 Jordan- blo ck ca se: ( λ x − f y ) ∂ ∂ x − f x ∂ ∂ y = 0 λ xx − 2 f xy = 0 (2( λU ) x − ( f 1 ) y ) ∂ ∂ x − ( f 1 ) x ∂ ∂ y = 0 2( λU ) xx − 2( f 1 ) xy = 0 and o bviously hold for λ , f and f 1 indicated a b ov e. The complex Liouv ille case is absolutely similar, the only difference is the additio nal term ∂ 2 ∂ y 2 , which leads to the following system o f relations: Complex Liouville case: ( λ x − f y ) ∂ ∂ x + ( − λ y − f x ) ∂ ∂ y = 0 λ xx − λ y y − 2 f xy = 0 (2( λU ) x − ( f 1 ) y ) ∂ ∂ x + ( − 2 ( λU ) y − ( f 1 ) x ) ∂ ∂ y = 0 2( λU ) xx − 2( λU ) y y − 2( f 1 ) xy = 0 each of whic h obviously holds for f = − ℜ ( h ) 2 , f 1 = −ℜ ( h 1 ), λ = ℑ ( h ) 2 , 2 λU = ℑ ( h 2 ). 3 Pro of of T heorem 1 3.1 Admissible co ordinate systems and Birkhoff-Kolok oltso v forms Let g b e a pse udo -Riemannian metric on M 2 of signature (+ , − ). Co ns ider (and fix) tw o vector fields V 1 , V 2 on M 2 such that • g ( V 1 , V 1 ) = g ( V 2 , V 2 ) = 0 and 11 • g ( V 1 , V 2 ) > 0. Such vector fields always e xist lo cally , (and since our r esult is lo cal, this is sufficient fo r our pr o of ). F or po ssible further use, let us note that s uch vector fields always exist on a finite (at most, 4-sheet-) cov er of M 2 . W e will say that a lo cal co o rdinate system ( x, y ) is admissi ble , if the vector fie lds ∂ ∂ x and ∂ ∂ y are prop ortiona l to V 1 , V 2 with po s itive co efficient of pro p o rtionality: ∂ ∂ x = λ 1 ( x, y ) V 1 ( x, y ) , ∂ ∂ y = λ 2 ( x, y ) V 2 ( x, y ) , where λ i > 0 . Obviously , • a dmissible co ordinates exist in a sufficiently s mall neigh b ourho o d of every p oint, • the metr ic g in admissible co or dinates has the fo r m ds 2 = f ( x, y ) dxdy , where f > 0 , (21) • tw o admissible co ordinate systems in one neig hbourho o d are c onnected by  x new y new  =  x new ( x old ) y new ( y old )  , where dx new dx old > 0, dy new dy old > 0 . (22) Lemma 1. L et ( x, y ) b e an admissible c o or dinate system for g . L et F given by (1) b e an inte gr al for g . Then, B 1 := 1 p | a ( x, y ) | dx, resp ectively , B 2 := 1 p | c ( x, y ) | dy ! is a 1-form, which is define d at p oints such that a 6 = 0 (r esp e ctively, c 6 = 0 ). Mor e over, the c o efficient a (r esp e ctively, c ) dep ends only on x ( re sp e ctively, y ), which in p articular implies that the forms B 1 , B 2 ar e close d. R emark 12 . The for ms B 1 , B 2 are not the direct analo g of the “Birkho ff” 2-form introduced b y Kolo koltso v in [22]. In a cer tain sense, they a re a r eal ana log o f the square r o ot of the Bir k hoff for m. Pro of of Lemma 1. The first part of the statement, namely tha t 1 p | a ( x, y ) | dx, resp ectively , 1 p | c ( x, y ) | dy ! transforms as a 1- form under admissible co o rdinate changes is evident: indeed, after the co or dinate change (22), the momenta tra ns form as follows: p x old = p x new dx new dx old , p x old = p x new dx new dx old . Then, the integral F in the new co ordinates has the for m  dx new dx old  2 a | {z } a new p 2 x new + dx new dx old dy new dy old b | {z } b new p x new p y new +  dy new d y old  2 c | {z } c new p 2 y new . Then, the formal expression 1 √ | a | dx old (resp ectively, 1 √ | c | dy old ) transforms into 1 p | a | dx old dx new dx new resp ectively , 1 p | c | dy old dy new dy new ! , which is precisely the transforma tio n law of 1-forms. 12 Let us prove that the co efficient a (resp ectively , c ) depends only on x (r esp ectively , y ), whic h in par ticular implies that the forms B 1 , B 2 are clo sed. If g is given by (21 ), its Hamiltonian is H = 2 p x p y f , and the co ndition { H , F } = 0 reads 0 =  2 p x p y f , ap 2 x + bp x p y + cp 2 y  = 2 f 2  p 3 x ( f a y ) + p 2 x p y ( f a x + f b y + 2 f x a + f y b ) + p y p 2 x ( f b x + f c y + f x b + 2 f y c ) + p 3 y ( c x f )  , i.e., is e quiv a le n t to the following system of PDE:        a y = 0 , f a x + f b y + 2 f x a + f y b = 0 , f b x + f c y + f x b + 2 f y c = 0 , c x = 0 . (23) Thu s, a = a ( x ), c = c ( y ), which is eq uiv alent to state that B 1 := 1 √ | a | dx and B 2 := 1 √ | c | dy are closed forms (assuming a 6 = 0 and c 6 = 0). R emark 13 . F o r further use let us formulate o ne more consequence o f equa tions (23): if a ≡ c ≡ 0 in a neighbourho o d of a p o int, then bf = co nst, implying F ≡ const · H in the neig hbourho o d. Assume a 6 = 0 (r esp ectively , c 6 = 0) at a p oint p 0 . F or every p 1 in a sma ll neighbourho o d U of p 0 consider x new := Z γ : [0 , 1 ] → U B 1 ,     resp ectively , y new := Z γ : [0 , 1 ] → U B 2     , (24) with γ (0) = p 0 , γ (1 ) = p 1 . Lo cally , in the admissible co ordina tes, the functions x new and y new are given by x new ( x ) = Z x x 0 1 p | a ( t ) | dt, y new ( y ) = Z y y 0 1 p | c ( t ) | dt . (25) The c o ordinates ( x new , y old ),  ( x old , y new ), ( x new , y new ), res pe ctively  are admissible. In these co o rdi- nates the forms B 1 , B 2 are given by dx new , dy new implying that a = c = ± 1 (more precisely: a new = sign( a old ), c new = sign( c old )). 3.2 Pro of of Theorem 1 W e assume that g on M 2 of signature (+,– ) admits a nontrivial quadratic int egra l F given by (1). Co nsider the (1 , 1)-tenso r G g iven by (2). In a neighbourho o d of almost every p oint, the Jor dan nor mal form of this (1 , 1)-tens or is one of the following: Case 1  λ 0 0 µ  , where λ, µ ∈ R . Case 2  λ + iµ 0 0 λ − iµ  , where λ, µ ∈ R . Case 3  λ 1 0 λ  , wher e λ ∈ R . Moreov er, in view of Rema rk 1 3, there exis ts a neighbour ho o d of almost every p o int such that λ 6 = µ in case 1 and µ 6 = 0 in case 2. In the a dmissible coo rdinates, up to mu ltiplication of F by − 1, ca se 1 is eq uiv ale nt to the condition ac > 0, ca se 2 is equiv alent to the condition ac < 0 and, fina lly , case 3 is equiv a lent to the condition ac = 0. W e now consider all three cases. 13 3.2.1 Case 1: ac > 0 . Without loss o f genera lit y we assume a > 0, c > 0. Cons ider the co or dinates (24). In these co ordinates a = 1, c = 1 and equations (23) hav e the following simple form.  ( f b ) y + 2 f x = 0 , ( f b ) x + 2 f y = 0 . (26) This system can b e so lved. Indeed, it is equiv alent to  ( f b + 2 f ) x + ( f b + 2 f ) y = 0 , ( f b − 2 f ) x − ( f b − 2 f ) y = 0 , (27) which after the (non-admissible) change of co ordinates x new = x + y , y new = x − y , has the form  ( f b + 2 f ) x = 0 , ( f b − 2 f ) y = 0 , (28) implying f b + 2 f = Y ( y ), f b − 2 f = X ( x ). Thus, f = Y ( y ) − X ( x ) 4 , b = 2 X ( x ) + Y ( y ) Y ( y ) − X ( x ) . Finally , in the new co ordina tes, the metric and the integral hav e (up to a p oss ible multiplication by a constant) the form ( X − Y )( dx 2 − dy 2 ) , (29) 1 2  p 2 x − X ( x )+ Y ( y ) X ( x ) − Y ( y ) ( p 2 x − p 2 y ) + p 2 y  = p 2 y X ( x ) − p 2 x Y ( y ) X ( x ) − Y ( y ) . (30) 3.2.2 Case 2: ac < 0 . Without loss o f genera lit y we ca n a ssume a > 0, c < 0. Cons ider the normal co o rdinates (24). In these co ordinates a = 1, c = − 1 and equations (23) hav e the following simple form.  ( f b ) y + 2 f x = 0 , ( f b ) x − 2 f y = 0 . (31) W e see that these equations are the Ca uchy-Riemann c o nditions for the co mplex-v alued function f b + 2 if . Thu s, for an appropria te holomo r phic function h = h ( x + i y ) w e hav e f b = ℜ ( h ), 2 f = ℑ ( h ). Finally , in a certain co o rdinate system, the metric and the integral are (up to po ssible multiplication by constants) ℑ ( h ) dxdy a nd p 2 x − p 2 y + 2 ℜ ( h ) ℑ ( h ) p x p y (32) 3.2.3 Case 3: ac = 0 . Without loss of gener ality w e can ass ume a > 0, c = 0. Co nsider a dmissible co ordinates x, y , such that x is the norma l co or dinate from (2 4). In these co ordinates a = 1, c = 0, and the equations (23) hav e the following simple form.  ( f b ) y + 2 f x = 0 , ( f b ) x = 0 . (33) This system ca n be solved. Indeed, the second equation implies f b = − Y ( y ). Substituting this in the first equation we obtain Y ′ = 2 f x implying f = x 2 Y ′ ( y ) + b Y ( y ) and b = − Y ( y ) x 2 Y ′ ( y ) + b Y ( y ) . 14 Finally , the metric and the integral are  b Y ( y ) + x 2 Y ′ ( y )  dxdy and p 2 x − Y ( y ) b Y ( y ) + x 2 Y ′ ( y ) p x p y . (34) Moreov er, by the change y new = β ( y old ), equations (34) will be simply tra nsformed to:  b Y ( y ) β ′ + x 2 Y ′ ( y )  dxdy and p 2 x − Y ( y ) b Y ( y ) β ′ + x 2 Y ′ ( y ) p x p y . (35) Thu s, by putting β ( y ) = R y y 0 1 b Y ( t ) dt , w e can make the metric a nd the in tegral to b e  1 + x 2 Y ′ ( y )  dxdy a nd p 2 x − Y ( y ) 1 + x 2 Y ′ ( y ) p x p y . Moreov er, after the co o rdinate change x new = x old 2 and multiplication of the metric by 1 2 , the metric and the in tegra l hav e the fo rm from Theo rem 1 (1 + xY ′ ( y )) dxdy and p 2 x − 2 Y ( y ) 1 + xY ′ ( y ) p x p y . (36) Theorem 1 is proved. R emark 14 . Let us note that if d Y 6 = 0, then w e can take Y as the co o rdinate y . The n, the metric and the int egra l (34) will hav e the for m (see a lso Rema rk 4)  ˜ Y ( y ) − x 2  dxdy and p 2 x + y ˜ Y ( y ) − x 2 p x p y . (37) 4 Conclusions W e hav e dis c ussed integrable geo desic flows of pseudo-Riema nnia n metrics o n 2 -dimensional manifolds co n- structing (Theo rem 1) lo ca l norma l forms of s uch metrics. The normal fo r ms ar e of three types: Liouville (the ana logous of the Riemannian case), Complex-Lio uville and Jorda n-blo ck. W e hav e shown tha t these metrics, in analo gy with the Riemannian case, admit geo desic a lly equiv alen t metrics, can b e used to co n- struct a large family of na tural systems admitting integrals quadra tic in mo men ta, that these natur al systems are integrable by quadratur es and that the in tegrability of such systems can be gener alized to the qua nt um setting. 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