A Coordinate-Free Construction for a Class of Integrable Hydrodynamic-Type Systems

A Co ordinate-F ree Construction for a Class of In tegrable Hydro dynamic-T yp e Systems Maciej B laszak a and Artur Sergyey ev b a Department of Ph ysi cs, A. Mickiewicz Uni v er sit y Um ultowsk a 85, 61-61 4 P o zna ´ n, Poland b Mathemat ical Insti tute, Si lesian U niv ersity in Opav a, Na Rybn ´ ı ˇ cku 1, 746 01 Opav a, Czech Republi c E-mai l: blas zakm@ amu. edu.pl and Art ur.Se rgye yev@math.slu.cz July 12, 2008 Using a (1,1)-te nsor L with zero Nijenhuis torsion and maximal p ossible num b er (equal to the n um b er of dep endent v ariables) of distinct, functionally in dep endent eigenv alues w e d efine, in a co ordinate-free fashion, the seed systems whic h are w eakly nonlinear semi-Hamilto nian systems of a sp ecial form, and an infin ite set of conserv ation la ws for the seed systems. The r ecipro cal transformations constructed from th ese conserv ation la ws yield a considerably larger class of h y dro dynamic-t yp e systems from the seed systems, and w e sh o w that these n ew systems are again defined in a coordinate-free manner, using the tensor L alone, and, moreo v er, are w eakly nonlinear an d semi-Hamiltonian, so their general solution can b e obtained b y means of the generalized ho d ograph metho d of Tsarev. In tro ductio n In the presen t pap er w e deal with the sy stems of first order quasi-linear PDEs of the form u t = A ( u ) u x , (1) where u = ( u 1 , . . . , u n ) T , A is an n × n mat rix, the sup erscript T ind icates the transp osed mat rix. The systems (1) are usually called h ydr o dynam ic-typ e systems or disp ersionless systems. More sp ecifically , w e shall restrict ourselv es to considering the systems (1) whic h are semi-Hamiltonian in the sense of Tsarev [23] and w eakly nonlinear 1 [10]. Although the c lass of w eakly nonlinear se mi-Hamiltonian (WNSH) system s w a s extens iv ely studied in the literature, see e.g. [6, 10, 12, 13 , 14, 17] and refe rences therein, the results obtained so far w ere mostly presen ted in the distinguished co o rdinates, the so- called Riemann in v arian ts. In particular, in these co ordinates w e hav e a complete description of WNSH h ydro dynamic-t yp e systems [10, 6] and, moreo v er, the general solution in implicit form for an y suc h system can b e found. What is more, any WN SH system written in the Riemann in v ariants can b e linearized using a suitably chosen recipro cal transformation [10] (see e.g. [15, 16, 1 9, 13] a nd references therein for a general theory o f reciprocal tra nsformations). Ho wev er, not m uc h is kno wn so far about ho w to construct or identify WNSH system s in a co ordinate- free fashion or construct recipro cal transforma tions for suc h systems written in arbitrary co ordina tes. 1 Note that weakly no nlinear sys tems are also k nown as line arly de gener at e , see e.g . [21, 11, 13, 6]. 1 Ev en though there exists [18] a co ordinate- free v ersion of conditions under whic h a given h ydro dyn- amic-t yp e system is w eakly nonlinear and semi-Hamiltonian, the conditio ns in question written in the co ordinate-free form are quite cum b ersome, and constructing any reasonably large classes of WNSH systems in arbitrary co or dinates using these conditions is a virtually imp ossible task even for lo w v alues of n except fo r the simplest cases of n = 2 and n = 3. As for the case of arbitrary n , some results w ere obtained in [6] for a sp ecial class of the WNSH systems, namely , the seed sys tems, see below for details. In t he presen t pap er w e construct in a co o rdinate-free fashion fairly extensiv e classes of WNSH systems from the so-called seed systems. W e star t with a (1,1)- tensor L with zero Nijenhuis torsion and maximal p ossible num b er (equal to the num b er of dep endent v aria bles) o f distinct, functionally indep enden t eigen v alues. Using this tensor w e define, in a co ordinate-free fashion, a class of WNSH h ydro dynamic-type syste ms whic h w e call the se e d systems , see Section 1 b elo w for details. W e then o bserv e that the seed systems po ssess infinitely many nontrivial conserv a tion la ws of a sp e- cial form tha t c an b e written in a c o or di nate-fr e e fashion . Note that ev en though any semi-Hamiltonian system ha s infinitely man y conserv ation la ws [23], in general the re is no wa y to write them dow n explicitly in arbitr ary co ordinates. Using the ab o v e sp ecial conserv ation la ws w e construct the r ecipro cal transformations (10) for the seed systems and sho w that these transformations yield new large classes (21) of WNSH h ydro dynamic- t yp e systems a priori written in a c o ordinate-free fashion. Finally , using the explicit form of the re sulting systems in the Riemann in v ariants, w e write do wn general solutions for t he systems in question using the tec hnique fro m [1 0, 6], see Section 3 b elow for details. It is imp ortant to stress that, a s shown in Section 2 b elow, for writing down the recipro cal trans- formations in question it suffices to kno w the tensor L alone. Th us, the co ordinate-fr ee construction of w eakly nonlinear semi-Hamiltonian h ydro dynamic-t yp e systems laid o ut in the presen t pap er w o rks for any (1,1)-tensor with zero Nijenh uis to rsion and maximal p ossible n um b er of distinct, functionally indep enden t eigenv alues. Moreo ve r, as sho wn in Section 1 b elow, this tensor alw ay s admits an infinite family of metrics for whic h it is an L -tensor in the sense of [3, 4, 5]. 1 The seed syst ems Consider an n -dimensional manifold M endo w ed with a tensor 2 L of t yp e (1,1 ), i.e., w ith one co v ariant and one con tra v arian t index, with zero Nije nh uis torsion and n distinct, functionally independent eigen v alues. It can b e sho wn that a tensor with these prop erties alw ays is an L -tensor [3, 4, 5], also kno wn as a sp ecial conformal Killing tensor of trace t yp e [9]. F o llo wing [3, 5, 7], consider the follo wing set of tensors of t yp e (1,1 ) on M : K 1 = I , K r = r − 1 X k =0 ρ k L r − 1 − k , r = 2 , . . . , n, (2) where I is the n × n unit matrix, and ρ i are co efficien ts of the characteristic p olynomial of the tensor L , i.e., det( ξ I − L ) = n X i =0 ρ i ξ n − i . (3) 2 F or the sake of br evity in wha t fo llows we sha ll use the term ‘tensor’ ins tead of ‘tenso r field’. 2 No w consider a vicinit y U ⊂ M with lo cal co o rdinates u 1 , . . . , u n , and a set of hydrodynamic-type systems of the form K − 1 1 u t 1 = K − 1 2 u t 2 = · · · = K − 1 n u t n (4) where let u = ( u 1 , . . . , u n ) T , and the superscript T refers to the matrix transp osition, t i are indep enden t v a riables, K − 1 i are tensors of t yp e (1,1) suc h that K i K − 1 i = I , i = 1 , . . . , n . F o r an y fixed j ∈ { 1 , . . . , n } w e can rewrite (4) as u t i = K i K − 1 j u t j , i = 1 , . . . , n, i 6 = j. (5) Notice that in (5) the v ariable t j pla ys the role of a space v ariable while the remaining times t i should b e considered as ev olution parameters. Moreov er K i K − 1 j again is a tensor of t yp e (1 , 1). It is imp o rtan t to stress that the set (4) (or (5)) o f h ydro dynamic-type sys tems is cov ariant under arbitrary c hanges of lo cal co ordinates on M , and in fact the systems in question are w ell-defined on the whole of M . W e shall refer to the systems (4) or (5) with K i giv en b y (2) as to the se e d systems . In fact, these systems b elong to a broader class of the so-called dispersionless Killing syste ms [8]. It can b e sho wn [6] that the seed sys t ems are w eakly no nlinear and semi-Hamiltonian. It is imm ediate fro m (2) that if w e c ho ose the eigen v alues λ i , i = 1 , . . . , n , of L fo r the lo cal coordina tes on U (this is p o ssible b ecause the Nijenh uis torsion of L v anishes a nd the eigenv alues in question are simple and functionally indep enden t), the quan tities (2) will b e diagonal in these co ordinates, and thus the eigen v alues in question will pro vide the Riemann in v arian ts for the seed systems (5). As will b e sho wn in Section 5, the solutio n for these systems, when expressed using the R iemann inv ariants , ha s the form (46). Not e that the Lax r epresen tations for these system s also app ear in the con text of the so-called univ ersal hierarch y [1, 2]. In terestingly enough, for the seed systems we hav e [8] an infinite set of conserv at ion laws tha t can be constructed in a c o or dinate-fr e e fashion . In order to write this se t do wn w e need t he so- called b asic se p ar a ble p otentials V ( k ) r that can b e defined using the tensor L via the following recursion relation [7 ]: V ( k ) r = V ( k − 1) r +1 − ρ r V ( k − 1) 1 , k ∈ Z , (6) with the initial condition V (0) r = − δ n r , r = 1 , . . . , n. (7) Here and b elo w w e tacitly assum e that V ( k ) r ≡ 0 for r < 1 or r > n . The recursion (6) can b e rev ersed. The in vers e recursion is given by V ( k ) r = V ( k +1) r − 1 − ρ r − 1 ρ n V ( k +1) n , k ∈ Z , r = 1 , . . . , n. (8) Hence, the first nonconstan t p otentials are V ( n ) r = ρ r for k > 0 and V ( − 1) r = ρ r − 1 ρ n for k < 0, resp ectiv ely . The conserv ation laws in question read [8] D t i ( V ( k ) j ) = D t j ( V ( k ) i ) , i, j = 1 , . . . , n, i 6 = j, k ∈ Z , (9) where D t i are total deriv atives computed b y virtue of (4). These conserv ation law s are ob viously non- trivial for all in teger k 6 = 0 , . . . , n − 1. 3 2 Recipro cal transformations and more Using (9) w e can define a large class of recipro cal transformations for the seed systems. Using these transformations w e construct extensiv e new classes (21) of WNSH hyd ro dynamic-ty p e systems. Most imp ortantly , these transformed systems, just lik e their seed counte rparts, possess an infinite set of non- trivial conserv ation laws t hat can b e construc ted in a co ordinate-free fashion, and the general solution of an y of the transformed systems (21) written in the Riemann in v arian t s tak es the form (47) and (48). The recipro cal transformatio n in question is defined for the whole set (4) of the seed systems and reads [20] as follo ws: d ˜ t s i = − n X j = 1 V ( γ i ) j dt j , i = 1 , . . . , k , ˜ t m = t m , m = 1 , 2 , . . . , n, m 6 = s a for an y a = 1 , . . . , k . (10) Here 1 ≤ k ≤ n ; the n um b ers s a , a = 1 , . . . , k , are a k - tuple of distinct integers fr om the set { 1 , . . . , n } , and γ j are arbitrary p ositiv e integers that satisfy the fo llo wing conditions: γ 1 > γ 2 > · · · > γ k > n − 1 . (11) The c hoice of num b ers k ∈ { 1 , . . . , n } , s a , and γ a that satisfy the ab ov e conditions uniquely determines the transformation (10). Using (9) w e can readily c hec k tha t (10) is a we ll-defined recipro cal transforma tion. The in ve rse of (10) has the form dt s i = − n X j = 1 ˜ V ( n − s i ) j d ˜ t j , i = 1 , . . . , k , t l = ˜ t l , q = 1 , 2 , . . . , n, l 6 = s a for an y a = 1 , . . . , k . (12) Here ˜ V ( m ) j are deforme d sep ar able p otentials defined for all in teger m as follo ws: 1) for j = s 1 , . . . , s k w e define ˜ V ( m ) s i b y means of the relations V ( m ) s i + k X p =1 ˜ V ( m ) s p V ( γ p ) s i = 0 , (13) whence ˜ V ( m ) s i = − det W ( m ) i / det W , (14) where W is a k × k matrix of the form W =               V ( γ 1 ) s 1 · · · V ( γ k ) s 1 . . . . . . . . . V ( γ 1 ) s k · · · V ( γ k ) s k               , (15) and W ( m ) i are obtained from W b y replacing V ( γ i ) s j b y V ( m ) s j for all j = 1 , . . . , k ; 2) for j 6 = s 1 , . . . , s k w e set ˜ V ( m ) j = V ( m ) j + k X p =1 ˜ V ( m ) s p V ( γ p ) j , (16) 4 or equiv alen t ly ˜ V ( m ) j = det ˆ W ( m ) j / det W , (17) where ˆ W ( m ) j is a ( k + 1) × ( k + 1) matrix of the form ˆ W ( m ) j =                     V ( m ) j V ( γ 1 ) j · · · V ( γ k ) j V ( m ) s 1 V ( γ 1 ) s 1 · · · V ( γ k ) s 1 . . . . . . . . . . . . V ( m ) s 1 V ( γ 1 ) s k · · · V ( γ k ) s k                     . (18) It can be show n that the ab o v e definition of ˜ V ( j ) i is equiv alen t to the one g iv en in [20]. In order to find out how Eq.(4) transforms under (1 0), w e temp orarily rewrite the former as u t i = K i Y , i = 1 , . . . , n, (19) where Y is an arbitrary v ector field on M . The transformation (10) sends the set (1 9) of seed systems into the f ollo wing set: u ˜ t i = ˜ K i Y , i = 1 , 2 , . . . , n, (20) whic h up o n eliminatio n of Y can b e written in the form similar to (4): ˜ K − 1 1 u ˜ t 1 = ˜ K − 1 2 u ˜ t 2 = · · · = ˜ K − 1 n u ˜ t n , (21) and can be further rewritten like (5) u ˜ t i = ˜ K i ˜ K − 1 j u ˜ t j , i = 1 , . . . , n, i 6 = j, (22) for an y fixed j ∈ { 1 , . . . , n } . As will b e sho wn in Section 5, t he general solution for these systems is giv en b y (47) and (48). Using (12) and the c hain rule w e find, after a straightforw ard but tedious computation, that ˜ K s i = − k P j = 1 ˜ V ( n − s j ) s i K s j M − 1 , i = 1 , . . . , k , ˜ K m = K m M − 1 − k P l =1 ˜ V ( n − s l ) m K s l M − 1 , m = 1 , 2 , . . . , n, m 6 = s a for an y a = 1 , . . . , k , (23) where M = − det W s 1 / det W, (24) W is giv en by (15), and W s 1 is o btained from W b y replacing V ( γ 1 ) s j b y K s j for a ll j = 1 , . . . , k . He re det W s 1 is a formal determinant with matrix-v alued en tries of the same kind as in [7]. Lik ewise, fro m (10) w e infer that K s i = − k P j = 1 V ( γ j ) s i ˜ K s j ˜ M − 1 , i = 1 , . . . , k , K m = ˜ K m ˜ M − 1 − k P l =1 V ( γ l ) m ˜ K s l ˜ M − 1 , m = 1 , 2 , . . . , n, m 6 = s a for an y a = 1 , . . . , k . (25) 5 Here ˜ M = − det f W s 1 / det f W , (26) f W is a k × k matrix of the form f W =               ˜ V ( γ 1 ) s 1 · · · ˜ V ( γ k ) s 1 . . . . . . . . . ˜ V ( γ 1 ) s k · · · ˜ V ( γ k ) s k               , (27) and f W s 1 is obtained from f W by replacing ˜ V ( γ 1 ) s j b y ˜ K s j for all j = 1 , . . . , k . Eq.(21) p ossesses the follo wing infinite set of non trivial conserv ation la ws similar to (9): D ˜ t i ( ˜ V ( m ) j ) = D ˜ t j ( ˜ V ( m ) i ) , i, j = 1 , . . . , n, i 6 = j, m ∈ Z , m 6 = γ l , l = 1 , . . . , k , m 6∈ ( { 1 , . . . , n } / { s 1 , . . . , s k } ) , (28) where the deriv ative s D ˜ t i are computed b y virtue of (21). As w e hav e already men tio ned ab o v e, an y tensor L of t yp e (1,1) with zero Nijenh uis to rsion and n distinct, functionally indep enden t eigen v alues alw ays is an L -tensor f or some fa mily of metrics on M . In fa ct, in the co ordinate frame asso ciated with the eigen v alues λ i , i = 1 , . . . , n , of L , the most general family of suc h contra v arian t metrics is giv en b y (44). The quan t ities K i (2) are then Killing tens ors of t yp e (1,1) for any metric tensor from the family (44). Using the results of [7] it can b e sho wn that the quantities ˜ K i are Killing tensors of t yp e (1,1) for a con trav ariant metric M G , where G is any contra v arian t metric from the family (4 4). Th us Eq.(21) (or equiv alen tly Eq.(22)) indeed defines a set of disp ersionless Killing system s, and the systems (22) are w eakly nonlinear and semi-Hamiltonian. Note that the w eak nonlinearit y of (21) can also b e inferred from the general result o f F erap onto v (Prop osition 3.2 of [11]) stating that recipro cal tr ansformations of h ydro dynamic-type systems preserv e w eak nonlinearit y . Alternativ ely , o ne can r eadily verify weak nonlinearit y and semi-Hamiltonicity o f (21) in the co o rdinate fr ame asso ciated with the eigen v alues λ i , i = 1 , . . . , n , of L . Moreov er, in the next section w e sho w how to construct a general solution for any system (21) in this co ordinate frame using the method fro m [1 0, 6]. 3 W e akly no nlinear s emi-Hamiltonian sys tems in Rie mann in- v arian ts: ge neral solutio n from se paration relation s Consider a h ydro dynamic-ty p e system written in the Riemann in v a rian ts: λ i t = v i ( λ ) λ i x , i = 1 , . . . , n, (29) where λ = ( λ 1 , . . . , λ n ), and there is no sum ov er i . The system (29) is said to b e we ak ly nonline ar (or line arly de gener ate , see e.g. [10, 21] and references therein) if ∂ v i /∂ λ i = 0 , i = 1 , . . . , n, (30) and is said to b e semi-Hamiltonian [23] if ∂ ∂ λ j  ∂ v i /∂ λ k v k − v i  = ∂ ∂ λ k  ∂ v i /∂ λ j v j − v i  , i, j, k = 1 , . . . , n, i 6 = j 6 = k 6 = i. (31) 6 It is natural to ask whic h is the most general w eakly nonlinear semi-Hamiltonian (WNSH) hydrody- namic-t yp e system (29) written in t he Riemann inv ariants, or, in o ther w ords, whic h is the most general form of v i that satisfy (30) and ( 31). It turns out [10, 6] that any WNSH hydrodynamic-type system (29) admits n − 1 comm uting flows of the same kind, so w e actually ha ve a set of comm uting WNSH hy dro dynamic-t yp e systems just lik e ( 5). In complete analogy with (5), this set can b e written in a symmetric form as λ i t 1 v i 1 = · · · = λ i t n v i n , i = 1 , . . . , n, (32) where v i 1 ≡ v i , i = 1 , . . . , n . The most general fo rm o f suc h a set of WNSH h ydro dynamic-ty p e syste ms is giv en by the formulas [10 , 6] v i r = ( − 1) r +1 det Φ ir det Φ i 1 . (33) Here Φ is a matrix of the form [10, 6] Φ =    Φ 1 1 ( λ 1 ) Φ 2 1 ( λ 1 ) · · · Φ n − 1 1 ( λ 1 ) Φ n 1 ( λ 1 ) . . . . . . · · · . . . . . . Φ 1 n ( λ n ) Φ 2 n ( λ n ) · · · Φ n − 1 n ( λ n ) Φ n n ( λ n )    , (3 4) where Φ i j ( λ i ) are arbitrary functions of the correspo nding v ariables; Φ ik is the ( n − 1 ) × ( n − 1) matrix obtained f rom Φ b y remo ving its i th ro w and k th column. Note that we can, without loss of generality , imp ose the normalization Φ n i = 1, i = 1 , . . . , n , but w e shall not use this normalization b elo w. The general solution for (32) can be written as [10, 6] n X j = 1 Z λ j Φ n − r j ( ξ ) ϕ j ( ξ ) dξ = t r , r = 1 , . . . , n, (35) where ϕ j ( ξ ) are a rbitrary functions of a single v ariable. If w e fix r , k ∈ { 1 , . . . , n } , r 6 = k , a nd consider t he sys tem λ i t k = v i k v i r λ i t r , i = 1 , . . . , n, (36) then the general solution of (36 ) is giv en b y (35) with t j = const fo r all j 6 = r , k . F o r an y pair ( r , k ) the system (36) represe n ts (29), where t k = t , t r = x , and v i = v i k /v i r satisfy the conditions (30) and (3 1). Note that to a giv en matrix Φ (34), or, equiv alen tly , to a set of n Killing tensors and a class of metrics that admit them, w e can asso ciate the so-called sep ar ation r elations of the form n X j = 1 Φ j i ( λ i ) H j = f i ( λ i ) µ 2 i , i = 1 , . . . , n, (37) where H j are separable geo desic Hamiltonians and f i ( ξ ), i = 1 , . . . , n , are arbitrary functions of a single v a riable [22, 6, 2 0] whic h are related to their coun terpar ts in (35) via the f orm ula ϕ i ( ξ ) = f i ( ξ ) n X j = 1 Φ j i ( ξ ) a j ! 1 / 2 , (38) 7 where a j are arbitrary constan ts. F ro m the p o in t of view of separation relations (37) the matrix Φ (34) is nothing but t he St¨ ac ke l matrix related to the Hamiltonians H i . Moreo ver, the comm utativity of H i implies the comm utativit y of the associated flows (3 2), and the general solution (35) for (32 ) in f act can b e obtained [10, 6] from the general solution for the simultaneous equations of motion for H i whic h, in turn, is found using the separation relations (37). Let us briefly recall the rationale behind the separation relations (3 7). If we define t he Hamiltonians H i = H i ( λ , µ ), i = 1 , . . . , n , where µ = ( µ 1 , . . . , µ n ) T , as solutions of the system (37) of linear algebraic equations then these Hamiltonians ha ve the fo rm H i = µ T K i G µ = n X r,s =1 µ r ( K i G ) r s µ s , i = 1 , . . . , n, (39) and a re w ell-known to Pois son comm ute with respect to the canonical Poiss on brack et { λ i , µ j } = δ j i . Quite naturally , K i are Killing tensors of t yp e (1 , 1) for a contra v arian t metric G . How ev er, it is imp ortant to stress that in general these K i do not neces sarily hav e the form (2). W e kno w from [20] that for the seed sys tems the separation relations (37) read n X j = 1 ( λ i ) n − j H j = f i ( λ i ) µ 2 i , i = 1 , . . . , n, (40) while for the syste ms (21) w e hav e k X j = 1 ( λ i ) γ j ˜ H s j + n X p =1 ,p 6 = s 1 ,...,s k ( λ i ) n − p ˜ H p = f i ( λ i ) µ 2 i , i = 1 , . . . , n. (41) Using (40) and (41) we can readily read off the functions Φ j i from (34) asso ciated with (4) for K i giv en b y ( 2), and with (21), a nd construct general solution fo r any give n seed system from (5), and for the transformed systems (22), b y the metho d of [10, 6], see b elo w for details. F o r the sp ecial case o f (40) the Killing tensors K i in (39) are given b y (2), and in the λ -co ordinates L has the form L = diag ( λ 1 , . . . , λ n ) . (42) On the other hand, from the separation relations (41) we find that ˜ H i = µ T ˜ K i ˜ G µ = n X r,s =1 µ r ( ˜ K i ˜ G ) r s µ s , i = 1 , . . . , n, (43) where ˜ K i are giv en b y (23) a nd ˜ G = M G with M give n b y (24 ). Th us, the Hamiltonian H i (resp. ˜ H i ) is naturally asso ciated with the t wice con tra v a rian t Killing tensor K i G (resp. ˜ K i ˜ G ), and vice v ersa. No w, the Hamiltonian H 1 asso ciated with K 1 G = G , i.e., with the original contra v arian t metric G itself, is the co efficien t at the highest p o w er of λ i on the left-hand side of (4 0). Like wise, in view of (11) the co efficien t at the highest p o w er of λ i on the left- hand side of (41) equals ˜ H s 1 . This is the r eason wh y it is natural to consider the con tr a v ariant metric ˜ G asso ciated with ˜ H s 1 from (41 ) as a natural coun terpart of the original contra v arian t metric G , cf. [7]. 8 Note also that the set of Hamiltonians H i , i = 1 , . . . , n , in (40) is related to the set of ˜ H i , i = 1 , . . . , n , in (41) via the so-called m ultiparameter generalized St¨ ac k el transform of a sp ecial form, see [20] for fur- ther details, and this v ery fact uniquely dete rmines the shap e of the recipro cal transformatio n (10) and (12) relating (5) and (21). Let us now apply the ab ov e results on general solutions to the seed systems (4) and their transformed coun terparts (21) in the co ordinates λ i b eing the eigen v alues of L . In the coordina tes in question (42) holds b y assumption. Note that an y metric G that admits L of the form (42) as an L -tensor in the λ -co o rdinates can b e written in the form [7 ] G = diag  f 1 ( λ 1 ) ∆ 1 , . . . , f n ( λ n ) ∆ n  , (44) where ∆ i = Q j 6 = i ( λ i − λ j ). The class (4 4) with arbitra ry functions f i ( ξ ) is precisely the class of the metrics that admit the set o f Killing tensors giv en b y (2) with L of the form (42). The join t general solution for the set of systems (4) written in the Riemann inv ariants , that is, λ i t 1 G ii ∂ ρ 1 /∂ λ i = · · · = λ i t n G ii ∂ ρ n /∂ λ i , i = 1 , . . . , n, or equiv alen t ly , λ i t 1 ∂ ρ 1 /∂ λ i = · · · = λ i t n ∂ ρ n /∂ λ i , i = 1 , . . . , n, (45) where w e used the form ula P r − 1 j = 0 ( λ i ) r − 1 − j ρ j = ∂ ρ r /∂ λ i (see e.g. [8]), reads n X j = 1 Z λ j ξ n − r ϕ j ( ξ ) dξ = t r , r = 1 , . . . , n. (46) Notice that ρ i are nothing but the Vi ` ete polynomials in the v ariables λ . Lik ewise, using (41) w e see that the general solution of (21) in implicit form reads n X j = 1 Z λ j ξ γ q ϕ j ( ξ ) dξ = ˜ t s q , q = 1 , . . . , k , (47) n X j = 1 Z λ j ξ n − i ϕ j ( ξ ) dξ = ˜ t i , i = 1 , . . . , n, i 6 = s q , q = 1 , . . . , k . (48) Ac kno wledgments This researc h w as supp orted in part b y the Ministry of Education, Y o uth and Sp orts of the Czec h Republic (M ˇ SMT ˇ CR) under gran t MSM 478130 5904, b y the Ministry of Science a nd Higher Education (MNiSW) of the R epublic o f P oland under the researc h gran t No. N N202 4049 33 , and b y Silesian Univ ersit y in Opa v a under grant IGS 9/2008. M.B. appreciates the w arm hospitalit y of the Mathematical Institute of Silesian Univers it y in Opa v a, where the presen t w ork w as initiated. A.S. is pleased to thank to the Departmen t of Ph ysics of the Adam Mick iewicz Unive rsit y in P ozna ´ n for the w arm hospitalit y extended to him at the p en ultimate stage of preparation of the presen t pa p er. 9 References [1] L. Mart ´ ınez Alonso, A.B. Sh abat, T o wards a theory of d ifferen tial constr ain ts of a hydrod ynamic hierarc hy . J. Nonlinear Math. Ph ys. 10 (2003), no. 2, 229–24 2; preprint nlin.SI/03100 36 [2] L. Mart ´ ınez Alonso, A.B. 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