On Initial Data in the Problem of Consistency on Cubic Lattices for $3 times 3$ Determinants

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 7 (2011), 075, 19 pages On Initial Data in the Problem of Consiste ncy on Cubic Lattices for 3 × 3 Determinan ts ⋆ Ole g I. MOKHOV †‡ † Centr e for Nonline ar Studies, L.D.L andau Institute for The or etic al Physics, R ussian A c ademy of Sc i enc es, 2 Kosygina Str., M osc ow, R ussia E-mail: mokhov@mi.r as.ru , mokhov@landau.ac.ru, mokhov@bk.ru ‡ Dep artment of Ge ometry and T op olo gy, F aculty of Me chanics and Mathematics, M.V. L omo nosov Mosc ow State University, Mosc ow, R ussia Received January 23, 2011, in f inal form July 17, 2011; P ublished o nline July 2 6, 2 011 doi:10.38 42/SIGMA.20 11.075 Abstract. The pap er is dev oted to complete pro ofs of theorems o n consistency on c ubic lattices for 3 × 3 deter minants. The discr ete nonlinea r equatio ns on Z 2 def ined by the condition that the determinants of a ll 3 × 3 matrices of v alues of the scala r f ield at the p oints of the lattice Z 2 that fo rm elementary 3 × 3 squares v anish are cons idered; some explicit concrete conditions of genera l po sition on initial data are formulated; and for a rbitrary initial data satisfying these co ncrete conditions o f genera l po sition, theor ems on co nsistency on cubic la ttices (a consis tency “a round a cub e” ) for the consider ed discrete nonlinea r equations on Z 2 def ined b y 3 × 3 determinants are prov ed. Key wor ds: consistency pr inciple; sq uare and cubic la ttices; integrable discrete equation; initial data ; determinant; bent elementary square; co nsistency “around a cub e” 2010 Mathematics Subje ct Classific ation: 39 A05; 52C0 7; 15A15; 37K10; 11H06 1 In tro duction In this p ap er w e presen t complete pro ofs of theorems on consistency on cubic lat tices for 3 × 3 determinan ts. F orm ulati ons and a scheme of p ro ofs of these theorems w ere giv en by the au thor in [1, 2], where a n ew, mo dif ied, consistency principle on cub ic lattices for a sp ec ial class of t w o-dimensional discrete equ ations def ined b y relations on elemen tary N × N squares of the square lattic e Z 2 , N > 2, w as pr op osed. Earlier, in [3, 4, 5], the remark able and v ery natural principle of c onsistency on cu bic lattic es w as p rop osed for discrete equations def ined by relations on elemen tary 2 × 2 squares of the square lattice Z 2 as an ef fectiv e test sin gling out a certain sp ecial class of “integ rable” discrete equations (see also [6, 7, 8 , 9 , 10, 11, 12, 13, 1 4]). In this pap er, we consider on ly the discrete nonlinear equations on Z 2 def ined by the condition that the determinan ts of all 3 × 3 matrices of v alues of the corresp onding scalar f ield u at the p oints of the lat tice Z 2 that form elemen tary 3 × 3 squares v anish. W e form ulate some explicit concrete conditions of general p osition on initial data, and for arb itrary initial data satisfying these concrete conditions of general p osition, w e pro v e theorems on consistency on cub ic lattices (a consistency “around a cub e”) for th e considered discrete nonlinear equations on Z 2 def ined b y 3 × 3 determinants. ⋆ This paper is a contribution to the Pro ceedin gs of th e Conference “Symmetries and Integrabili ty of Dif ference Equations (SID E- 9)” (Jun e 14–18, 2010, V arna, Bulgaria). The full collection is av ailable at http://w ww.emis.de/j ournals/SIGMA/SIDE-9.html 2 O.I. Mokhov 2 Consistency p rinciple on cubic lattices W e consider th e square lattice Z 2 consisting of all p oin ts with arbitrary in teger co ordin ates in R 2 = { ( x 1 , x 2 ) | x k ∈ R , k = 1 , 2 } and complex (or real) scalar f ields u on the lattice Z 2 , u : Z 2 → C , d ef ined by their v alues u i 1 i 2 , u i 1 i 2 ∈ C , at eac h lattice p oint w ith co ordinates ( i 1 , i 2 ), i k ∈ Z , k = 1 , 2. W e consider a class of tw o-dimensional discrete equations on the lattice Z 2 for the f ie ld u that are giv en b y fun ctions Q ( x 1 , x 2 , x 3 , x 4 ) of four v ariables with the help of the relations Q ( u ij , u i +1 ,j , u i,j +1 , u i +1 ,j +1 ) = 0 , i, j ∈ Z , (1) so that in eac h elementary 2 × 2 squar e of the lattic e Z 2 , i.e., in eac h set of lattice p oin ts with co ordinates of the form { ( i, j ) , ( i + 1 , j ) , ( i, j + 1) , ( i + 1 , j + 1) } , i, j ∈ Z , the v alue of the f ield u at one of the ve rtices of the square is determined b y the v alues of the f ield at the other three v ertices. In this case th e s calar f ield u on the lattice Z 2 is completely determined by f ixin g in itial data, for examp le, on the coordin ate axes of the lattice, u i 0 and u 0 j , i, j ∈ Z . Here, w e do not discuss conditions on the initial data u ij themselv es that m ust correctly and completely d etermine a scalar f ield u on the lattice Z 2 for concrete discr ete equations of the form (1), a nd also w e d o not discuss conditions on the functions Q ( x 1 , x 2 , x 3 , x 4 ) for whic h relations (1) correctly determine a t w o-dimensional d iscrete equation on the lattice Z 2 for the f ield u . W e consider the cubic lattice Z 3 consisting of p oints with intege r co ordinates in R 3 = { ( x 1 , x 2 , x 3 ) | x k ∈ R , k = 1 , 2 , 3 } and f ix initial dat a, for example, on the co ordinate axes of the lattice, u i 00 , u 0 j 0 , and u 00 k , i, j, k ∈ Z . A t w o-dimensional discrete equation (1) is said to b e c onsistent on the cu bic lattic e (see [3, 4, 5, 6 , 7]) if for generic initial data the discrete equation (1 ) can b e satisf ied in a consisten t w a y simultaneously on all tw o- dimensional c o or dinate su blattices of the cubic lattice Z 3 that are def ined by f ixin g one of coordinates ( an y of the three coordinates) o f th e cubic lattice. This condition is equiv alen t to the consistency condition on eac h elementary 2 × 2 × 2 c ub e of the lattic e Z 3 , { ( i + p, j + r , k + s ) , 0 ≤ p, r, s ≤ 1 } , wh ere i, j, and k are arb itrary f ixed in tegers, i, j, k ∈ Z , i.e., relation (1) m ust b e satisf ied in a consisten t wa y on all f aces of an y elemen tary 2 × 2 × 2 cu b e of the lattice Z 3 for generic in itial d ata. In th e elemen tary cub e { ( i, j, k ) , 0 ≤ i, j, k ≤ 1 } the v alues u 101 , u 110 , and u 011 are d etermined by relations (1) on the corresp ondin g faces of the cub e b y the initial data u 000 , u 100 , u 010 , and u 001 , and thr ee relations on three faces of the cub e must b e satisf ied for the v alue u 111 . On e can consid er the condition of consistency of the o v erd etermined system of relations for the v alue u 111 for generic initial data as the consistency condition for the discrete equation (1) on the cubic lattice Z 3 . Here, we do not discuss all v arious situations in whic h relations (1) correctly def ine a t wo- dimensional discrete equation for the f ield u on an y tw o-dimens ional co ord inate sublattice of the cubic latti ce Z 3 ; for example, one can assume for simplicit y that relatio ns (1) are in v arian t with resp ect to the full symmetry group of the square. Classif ications of discrete equations of the form (1) that are consistent on the cubic lattice were s tudied in [6] and [10] u nder some additional co nditions on the functions Q ( x 1 , x 2 , x 3 , x 4 ) (see also [11]). The equati on u i,j +1 u i +1 ,j − u i +1 ,j +1 u ij = 0 , i, j ∈ Z , (2) def ined by the condition that the determinant s of all 2 × 2 matrices of v alues of the f ield u at the vertices of elemen tary 2 × 2 s quares of the lattice Z 2 v anish, is an example of suc h a tw o- dimensional nonlin ear discrete equation that is consistent on the cubic lattice. Equ ation (2) is linear w ith resp ect to eac h v ariable and inv ariant with resp ect to the full s ymmetry group of On Initial Data in th e Problem of Consistency on Cubic Latti ces for 3 × 3 Determinan ts 3 the square. Fixing arbitrary n onzero initial data u i 0 and u 0 j , i, j ∈ Z , on the co ordin ate axes of the lattice Z 2 completely determin es a f ield u on the la ttice Z 2 satisfying the discrete nonlinear equation (2), and f ixing arbitrary nonzero in itial data u i 00 , u 0 j 0 , and u 00 k , i, j, k ∈ Z , on th e co ordinate axes of the lattice Z 3 completely determines a f ield u on th e lattice Z 3 satisfying the discrete n onlinear equation (2) on all tw o-dimensional co ordinate sublattices of the cubic lattice Z 3 ; i.e., relations (2) are satisf ied in a consisten t wa y on all faces of eac h elemen tary 2 × 2 × 2 cub e of the lattice Z 3 for arb itrary nonzero initial data. The int egrabilit y (in the b roadest s ense of the w ord) of the discrete nonlin ear equation (2) is obvio us, since it can b e easily linearized: ln u i,j +1 + ln u i +1 ,j − ln u i +1 ,j +1 − ln u ij = 0, i, j ∈ Z . Discrete nonlinear equations def ined b y determinan ts of higher orders are also C -in tegrable and it is not a problem to write their general solution for generic in itial data, but in an y case they are muc h more complicated, nonlinearizable, and the problem on their consistency on cubic lattice is v ery non trivial (see [1, 2]). 3 Relations on elemen ta ry 3 × 3 squares of the lattice Z 2 and consistency conditions W e will use the follo wing def in ition ev erywhere in this pap er. An e lementary N × N squar e of the squar e lattic e Z 2 is a set of p oints of the lattice Z 2 with co ordin ates { ( i + s, j + r ) , 0 ≤ s, r ≤ N − 1 } , where i, j is an arbitrary f ixed pair of int egers, i, j ∈ Z , N ≥ 2. Let u s consider a discrete equation on Z 2 def ined b y a relation for the v alues of the f ie ld u at th e p oin ts of the lattice Z 2 that form elementary 3 × 3 squar es : Q ( u ij , . . . , u i + s,j + r , . . . , u i +2 ,j +2 ) = 0 , 0 ≤ s, r ≤ 2 , i, j ∈ Z , (3) so that in eac h elemen tary 3 × 3 square of the latt ice Z 2 , i.e., in eac h set of lattic e p oint s with co ordinates of the form { ( i, j ) , ( i + 1 , j ) , ( i + 2 , j ) , ( i, j + 1) , ( i + 1 , j + 1) , ( i + 2 , j + 1) , ( i, j + 2), ( i + 1 , j + 2) , ( i + 2 , j + 2) } , i, j ∈ Z , the v alue of th e f ield u at one of the p oints of this elemen ta ry 3 × 3 squ are is determined b y the v alues of the f ield at the other eig h t p oint s. F or def in iteness, one can require, for example, that relations (3) are in v arian t with resp ect to the full symm etry group of the conf iguration of p oints of the lattice Z 2 that form elemen tary 3 × 3 s quares (obvio usly , this group of symmetries coincides with the full symmetry group of the usual square). F or an y d iscrete equation of the form (3), f ixing generic initial data, for example, on t w o bands along the coord inate axes of the lattice Z 2 , { ( i, 0) , ( i, 1) , i ∈ Z } and { (0 , j ) , (1 , j ) , j ∈ Z } , completely determines a f ield u on Z 2 that sati sf ies this equation. Quite similarly , discrete equations on Z 2 giv en by relations for the v alues of the f ield u at the p oint s of the lattice Z 2 that f orm element ary N × N squares can b e def ined for an arbitrary N ≥ 2. F or def initeness, one can again requir e, for example, that the relations are inv arian t with resp ect to the full symmetry group of the conf iguration of p oin ts of the lattice Z 2 that f orm elemen tary N × N s quares (moreo v er, it is also ob vious that for an y N ≥ 2 the full symmetry group of the conf iguration of p oint s of the lattice Z 2 that form elementa ry N × N s quares coincides with the full sy mmetry group of the us ual squ are). W e consider the cubic lattice Z 3 and the c onsistency c ondition for discr ete e quations of the form (3) on al l two-dimensional c o or dinate sublattic es of the cubic lattic e Z 3 . Initial data can b e sp ecif ied, for example, at the follo wing lattice p oin ts that are situated on 12 s traigh t lines going along the co ordinate axes of the lattice Z 3 : ( i, 0 , 0), ( i, 1 , 0), ( i, 0 , 1), ( i, 1 , 1), (0 , j, 0), (1 , j, 0), (0 , j, 1), (1 , j, 1), (0 , 0 , k ), (1 , 0 , k ), (0 , 1 , k ), and (1 , 1 , k ), i, j, k ∈ Z . The v alues of the f iel d u at all other p oints of the cubic lattice Z 3 m ust then b e correctly determined in a consisten t w a y b y relations (3) on all elemen tary 3 × 3 squ ares of all t w o-dimensional co ordinate sublattices of the cubic lattice Z 3 for generic initial d ata. In the elemen tary cub e { ( i, j, k ) , 0 ≤ i, j, k ≤ 2 } the v alues u 202 , u 212 , u 220 , u 221 , u 022 , and u 122 are determined using relations (3) on the 4 O.I. Mokhov corresp ondin g elemen tary 3 × 3 s quares situated in the cub e u nder consideration (three faces of the cub e that are situated on the coordinate planes and three middle normal sect ions of the cub e) by the initial data u 000 , u 100 , u 200 , u 010 , u 110 , u 210 , u 001 , u 101 , u 201 , u 011 , u 111 , u 211 , u 020 , u 120 , u 021 , u 121 , u 002 , u 102 , u 012 , and u 112 , and three relatio ns must hold simultaneo usly on three other faces of the cu b e for the v alue u 222 . One can assu me that the consistency cond ition on the cub ic lattice Z 3 for any discrete equation of the form (3) is the consistency condition of the corresp ondin g o v erdetermined system of relations on th e v alue u 222 for generic initial data. Quite similarly , for an arbitrary N ≥ 2, one can def ine the c onsistency c on dition on the cubic lattic e Z 3 for discr e te e quations on Z 2 given by r elat ions on the v alues of the field u at the p oints of the lattic e Z 2 that form elementary N × N squ ar es . The v alues of the f ield u at all p oin ts of the cub ic lattice Z 3 m ust b e correctly determined in a consistent w a y by relations on all elemen tary N × N squares of all t wo-dimensional coord inate sublattices of the cub ic lat tice Z 3 for generic initial data. The consistency of discrete equations on the lattice Z 2 that are giv en by relations on the v alues of the f ield u at the p oin ts of the lattice Z 2 that form elemen tary N × N squares can b e considered on one elemen tary N × N × N cub e (the consistency of the relat ions on all faces and all normal sections of the cub e th at are parallel to the co ordin ate planes for generic in itial data sp ecif ied in the cub e). Let us consider the discrete non linear equation on Z 2 def ined b y th e condition that the determinan ts of all 3 × 3 matrices of v alues of the f ield u at the p oin ts of th e lattice Z 2 that form ele men tary 3 × 3 squares v anish: u i,j +2 u i +1 ,j +1 u i +2 ,j + u i,j +1 u i +1 ,j u i +2 ,j +2 + u i,j u i +1 ,j +2 u i +2 ,j +1 − u i,j u i +1 ,j +1 u i +2 ,j +2 − u i,j +2 u i +1 ,j u i +2 ,j +1 − u i,j +1 u i +1 ,j +2 u i +2 ,j = 0 , i, j ∈ Z . (4) Equation (4) is linear with resp ect to eac h v ariable and in v arian t with resp ect to the full sym- metry group of the conf igur ation of p oin ts of the lattice Z 2 that form elementa ry 3 × 3 squares. It is not d if f icult to c h ec k that for generic initial data the ab o v e-co nsidered consistency condition on th e cubic lattice is n ot satisf ied for the discrete equation (4), and, in this sense, the discrete nonlinear equation (4) is not consisten t on t w o- dimensional co ordinate sublattices of the cubic lattice Z 3 . W e n ote that for suc h setting of th e consistency problem on the cubic lattice for discrete equations of the form (3) we ha v e the f ollo w ing: in the elemen tary 3 × 3 × 3 cub e { ( i, j, k ) , 0 ≤ i, j, k ≤ 2 } of the lattice Z 3 , giv en initial data of v alues of the f ield u at 20 p oint s of this elemen tary 3 × 3 × 3 cu b e, one ca n determine the v alues of the f ield u at the other sev en p oints of this elemen tary 3 × 3 × 3 cub e by relations (3 ) (nine relations on six f aces and on thr ee middle normal sections of the cub e), and only for the v al ue of the f ield at one of the p oints one obtains an o v erdetermined system consisting of three relations on three distinct element ary 3 × 3 squ ares. 4 Ben t elemen tary 3 × 3 squares and mo dif ied consistency conditions W e consider a d iscrete equation of the form (3) and require that the discrete equation is sa- tisf ied not only on all t w o-dimensional co ordinate sub lattices of the cubic lattice Z 3 , but also on all unions of t w o arb itrary in tersecting t w o-dimens ional co ordin ate s ublattices of the cubic lattice Z 3 ; i.e., th e corresp onding elemen tary 3 × 3 squares on wh ic h the discrete equation of the form (3) is considered can b e b ent at a righ t angle along any of t w o middle lines of the elemen tary 3 × 3 square passing from one tw o- dimensional co ord inate su blattice to another so that all p oin ts of b ent elementary 3 × 3 squar es are situated at lattice p oin ts (one of lines of any b ent elementa ry 3 × 3 square is situated on one of tw o in tersecting tw o-dimens ional co ordinate sub lattice s of the cubic latti ce Z 3 , one of the lines is situated on th e seco nd of these t w o sublattice s, and the On Initial Data in th e Problem of Consistency on Cubic Latti ces for 3 × 3 Determinan ts 5 middle line whic h this element ary square is b ent along is situated on the in tersection of th ese t w o in tersecting tw o -dimensional co ordin ate sub lattices), for example, { ( i, 0 , 0 ) , ( i, 1 , 0) , ( i, 0 , 1) , i = 0 , 1 , 2 } , { (0 , j, 0) , (1 , j, 0) , (0 , j , 1) , j = 0 , 1 , 2 } , and { (0 , 0 , k ) , (1 , 0 , k ) , (0 , 1 , k ) , k = 0 , 1 , 2 } ( b ent elementary 3 × 3 squar es ). W e will consider relation (3) on all elemen tary 3 × 3 squ ares (b en t and un b ent) all p oin ts of whic h are situated at lattice p oin ts. In this case initial d ata can b e sp ecif ied, for example, at the follo wing p oin ts of the cubic lattice Z 3 : ( i , 0 , 0), ( i, 0 , 1), (0 , j, 0), (0 , j, 1), (0 , 0 , k ), (1 , 0 , k ), (1 , 1 , 0), and ( 1 , 1 , 1), i, j, k ∈ Z . The v alues of the f ie ld u at all other points of the cubic lattice Z 3 m ust then b e co rrectly determined in a consistent w a y b y relations (3) on all elemen tary 3 × 3 squares (including all b ent elemen tary 3 × 3 squares) of all unions of t w o arbitrary in tersecting tw o-dimensional co ord inate sublattices of the cubic lattice Z 3 for generic initial d ata. In the elemen tary cub e { ( i, j, k ) , 0 ≤ i, j, k ≤ 2 } the v alues u 012 , u 022 , u 112 , u 202 , u 12 k , u 21 k , and u 22 k , 0 ≤ k ≤ 2, are determined b y the initial data u 000 , u 100 , u 200 , u 001 , u 110 , u 010 , u 101 , u 201 , u 111 , u 011 , u 002 , u 020 , u 021 , and u 102 and b y ov erd etermined systems generated by relations (3) on elemen tary 3 × 3 squares (includin g all b ent elemen tary 3 × 3 squares) that are situated in the cub e un der consideration (six faces, three m iddle normal sections of the cub e, and 48 b en t elemen tary 3 × 3 squares). T here are 48 distinct b ent elemen tary 3 × 3 squares in an y elemen tary 3 × 3 × 3 cub e, which can b e easily coun ted b y th e b endin g edges of th e b ent elementa ry 3 × 3 squares: to eac h of th e 12 edges of th e cub e, there corresp onds one b ent elemen tary 3 × 3 s quare in the cub e; to eac h of the 12 mid dle lin es of p oint s on the faces of the cub e (2 × 6), there corresp ond t w o distin ct b en t elemen tary 3 × 3 squ ares in the cub e; and to eac h o f t he three in terior lines o f points in the cub e that conn ect the cen tres of opp osite face s o f the cub e, there corresp ond four distinct b en t elemen tary 3 × 3 squares in the cub e. One can assume that the consistency condition on the cub ic lattice Z 3 for any discrete equ ation of the form (3 ) (the global consistency) is the consistency condition of the corresp ondin g ov erd etermined system of relations on the v alues of the f ie ld u in the elemen tary cub e { ( i, j, k ) , 0 ≤ i, j, k ≤ 2 } for ge neric initia l d ata (the lo cal co nsistency). The co rresp on ding discrete equatio ns will also b e called c onsistent on the cubic lattic e . W e note that f or this new setting of the consistency problem on th e cubic lattice for discrete equations of the form (3) we ha v e the f ollo w ing: in the elemen tary 3 × 3 × 3 cub e { ( i, j, k ) , 0 ≤ i, j, k ≤ 2 } of the lattice Z 3 , giv en initial data of v alues of the f ield u at 14 p oint s of this elemen tary 3 × 3 × 3 cu b e, one can determine the v alues of the f ield u at the other 13 p oint s of this element ary 3 × 3 × 3 cub e by relatio ns (3) (57 relations on six faces, on three mid dle normal sections of the cub e, and on 48 b en t elemen tary 3 × 3 squares), wh ic h constitute in this case a highly o v erdetermined system of relations. W e also note that the global consistency follo ws from the lo cal consistency , i.e., if the consis- tency condition is fulf ille d for generic initial d ata in th e elemen tary cu b e { ( i, j, k ) , 0 ≤ i, j, k ≤ 2 } , then it is fulf illed for generic in itial d ata everywhere in the cubic lattice Z 3 , i.e., in eac h ele- men tary 3 × 3 × 3 cub e { ( i 0 + i, j 0 + j, k 0 + k ) , 0 ≤ i, j, k ≤ 2 } , i 0 , j 0 , k 0 ∈ Z . W e giv e h ere a formal sc heme of pr o of. The n umber of give n initial d ata in an arbitrary elemen tary 3 × 3 × 3 cub e { ( i 0 + i, j 0 + j, k 0 + k ) , 0 ≤ i, j, k ≤ 2 } , i 0 , j 0 , k 0 ∈ Z , is not more th an 14 as w e hav e in th e “main” elemen tary cub e { ( i, j, k ) , 0 ≤ i, j, k ≤ 2 } . W e start from the “main” elementa ry cub e { ( i, j, k ) , 0 ≤ i, j, k ≤ 2 } , where the consistency cond ition is fulf illed for generic initial d ata as th e lo cal consistency b y our assu mption, and th en w e will consider some sp ecial shifts of elemen tary 3 × 3 × 3 cub es step by step from the “ma in” elementa ry cub e { ( i, j, k ) , 0 ≤ i, j, k ≤ 2 } un til f ill all the cubic lattice Z 3 . Th ere are thr ee essen tially d if ferent situatio ns, when after shifting we obtain new 9 p oin ts of the cubic lat tice Z 3 (the face of the shifted elemen ta ry 3 × 3 × 3 cub e), new 3 p oin ts (the edge of the sh ifted element ary 3 × 3 × 3 cub e) or new only one p oin t (the v ertice of the shifted elemen tary 3 × 3 × 3 cub e). First of all, w e co nsider 6 shifted elemen tary 3 × 3 × 3 cu b es (6 elemen tary 3 × 3 × 3 cub es shifted to the sid e of eac h of the faces of the “main” elemen tary cub e { ( i, j, k ) , 0 ≤ i, j, k ≤ 2 } ). In these elementa ry 3 × 3 × 3 cub es w e 6 O.I. Mokhov ha v e some giv en initial data (not more than 14) and some v alues of the f iel d u determined in the “main” elemen tary cub e { ( i, j, k ) , 0 ≤ i, j, k ≤ 2 } . Let us consider an y of these elementa ry 3 × 3 × 3 cub es and pr o v e that the consistency condition is also f ulf illed in it. W e consider some determined v alues as new initial data for this elemen ta ry 3 × 3 × 3 cub e, f irst of all, w e tak e giv en initial data and if it is n ecessary (i.e., if the n um b er of giv en initial d ata in this elemen tary 3 × 3 × 3 cub e is less than 14), add to give n initial data the corresp ondin g v alues d etermined in the “main” elemen ta ry cub e { ( i, j, k ) , 0 ≤ i, j, k ≤ 2 } . It is simple to c hec k that we alw a ys can do this. Then we can determine all v alues in this element ary 3 × 3 × 3 cub e in a consistent w a y by our assu mption of the lo cal consistency . After that we must pro v e that the v alues deter- mined simulta neously fr om the “main” elemen tary cub e { ( i, j, k ) , 0 ≤ i, j, k ≤ 2 } and f rom the elemen tary 3 × 3 × 3 cub e und er consideration coincide. It follo ws from the fact that all these v alues can b e determined step by step f rom elemen tary 3 × 3 squares (b ent and unbent) situated in b oth these el emen tary 3 × 3 × 3 cub es sim ultaneously . Similarly , w e pro v e step by s tep that the consistency condition is fulf ill ed in any elemen tary 3 × 3 × 3 cub e obtained from the “main” elemen tary cub e { ( i, j, k ) , 0 ≤ i, j, k ≤ 2 } by a shift along the co ordinate axes (along the given bands of initial d ata). Then w e pro v e s tep by step th at the consistency condition is f ulf illed in any elemen ta ry 3 × 3 × 3 cub e obtained fr om the consid ered elemen tary 3 × 3 × 3 cub es by shifts in the corresp ondin g co ordinate planes. This is the second t y p e of our sh ifts. Again w e m ust consider some initial data in eac h elementa ry 3 × 3 × 3 cu b e and determine all v alues in it in a consistent w a y by our assum ption of the lo cal consistency . Al l the v alues d etermined sim ultaneously from dif feren t elemen tary 3 × 3 × 3 cub es coincide s ince all these v alues can be determined step by step from elementa ry 3 × 3 squares (b ent an d unb en t) situated in b oth these elemen tary 3 × 3 × 3 cub es simulta neously . Similarly , we pro v e step by step that the consistency condition is fu lf illed f or th e shifts of the thir d type f illing all the cubic lat tice Z 3 . Quite similarly , for an arbitrary N > 2, on e can def ine the corresp ond ing mo difie d c onsistency c ondition on the cubic lattic e Z 3 for discr ete e q uations on the squ ar e lattic e Z 2 that ar e given by r elations on the values of the field u at the p oints of the lattic e Z 2 that form elementary N × N squar es ( including any b ent elementary N × N squar es ). Moreo v er, for N > 3, one can, generally sp eaking, all o w a larger num b er (up to N − 2) of b end ings of elemen tary N × N squ ares in the cubic lattic e Z 3 (in this case eac h elemen tary N × N squ are can b e b ent in the cubic lattice sim ultaneously along up to N − 2 parallel lines of the same t ype, eac h b end ing b eing to one of the t wo p ossible dif f eren t sides). 5 Consistency on cubic lat tices for 3 × 3 determinan ts The follo wing basic theorem holds . Theorem 1 ([1]) . F or arbitr ary generic initial data, the nonline ar discr ete e quation (4) c an b e satisfie d in a c onsistent way on al l unions of p airs of arbitr ary interse cting two-dimensional c o or dinate sublattic e s of the cubic lattic e Z 3 ; i.e., the discr ete nonline ar e quation (4) is c onsistent on the cubic lattic e Z 3 . Pro of . Here we giv e the strict pro of of consistency “around the element ary cub e” (the local consistency) f or generic initial data. Let us consider the elemen tary cub e { ( i, j, k ) , 0 ≤ i, j, k ≤ 2 } of the cubic la ttice Z 3 and sp ecify ge neric in itial data at th e follo wing p oint s of th is elemen tary cub e: ( i, 0 , 0), ( i, 0 , 1), (0 , j, 0), (0 , j, 1), (0 , 0 , k ), (1 , 0 , k ), (1 , 1 , 0), and (1 , 1 , 1), 0 ≤ i, j, k ≤ 2. In particular, it is suf f icien t to r equire that the follo wing condition on initial data is fu lf illed: in all elemen tary 3 × 3 squares (b en t and un b ent) situated in the elemen tary cub e u nder consideration, all 2 × 2 minors that are completely f ormed by only initial data are not equal to zero (this is a cond ition of general p osition for initial data). In this p ap er, w e will assume that precisely th is concrete condition of general p osition on initial data is f ulf illed. W e note that the cond ition On Initial Data in th e Problem of Consistency on Cubic Latti ces for 3 × 3 Determinan ts 7 on initial data, when no four v al ues u a , u b , u c , u d of initial data fr om distinct p oints a , b , c , d of the cub e under consideration s atisfy the relation u a u b = u c u d (i.e., if w e arrange v al ues of in itial data from distinct p oints of the cub e u nder consideration in all p oints of elemen ta ry 2 × 2 sq uare in an arbitrary w a y , then all the determinan ts of the corresp onding 2 × 2 matrices obtained b y this pro cedur e are not equal to zero), is also a condition of general p osition for initial data. W e require the fulf illment of a weak er condition on initial data, w hen only some of these determinan ts, namely , only those from them that are 2 × 2 minors formed by initial data in the 3 × 3 matrices of v alues of the f ie ld at the p oint s of the elemen tary 3 × 3 squares (b ent and u n b ent) situated in the elemen tary cub e u nder consideration, must not b e equal to zero. W e w ill distinguish the follo wing three dif f eren t t yp es of lines of lattice p oint s in the elemen tary cub e u nder consideration: the lines p arallel to th e x -axis, i.e., th e sets of p oint s of the form { ( i, r , s ) , 0 ≤ i ≤ 2 } , where ( r, s ) are f ixed ordered pairs of in tege rs, 0 ≤ r, s ≤ 2, that n um b er the lines in th is elemen tary cub e th at are parallel to the x -axis ( the x -typ e lines ); the lines parallel to th e y -axis, i.e., the sets of p oin ts of the form { ( r , j, s ) , 0 ≤ j ≤ 2 } , where ( r , s ) are f ixed ordered pairs of integ ers, 0 ≤ r , s ≤ 2, that num b er the lines in the elemen tary cub e under consideration that are parallel to the y -axis ( the y -typ e lines ); and the lines parallel to the z -axis, i.e., the sets of p oin ts of the form { ( r , s, k ) , 0 ≤ k ≤ 2 } , where ( r , s ) are f ixed ordered pairs of integ ers, 0 ≤ r , s ≤ 2, that num b er the lines in the elementa ry cub e under consideration that are paralle l to the z -axis ( the z -typ e lines ). The sp ecif ied in itial data f ill a pair of lines of eac h of these three t yp es (a pair of lines parallel to the corresp ondin g co ordinate axis for eac h of the co ord inate axes). W e will consider all these lines as b asic ones: { ( i, 0 , 0) , 0 ≤ i ≤ 2 } and { ( i, 0 , 1) , 0 ≤ i ≤ 2 } ( the b asic x -typ e lines ); { (0 , j, 0) , 0 ≤ j ≤ 2 } and { (0 , j , 1) , 0 ≤ j ≤ 2 } ( the b asic y -typ e lines ); and { (0 , 0 , k ) , 0 ≤ k ≤ 2 } and { (1 , 0 , k ) , 0 ≤ k ≤ 2 } ( the b asic z -typ e lines ). The v ectors of v alues of th e f ield u at the p oin ts of the basic lines will b e calle d b asic ve ctors (of the corresp onding t yp e). Note that the v ectors of v alues of th e f ield u at th e p oin ts of the basic lines of eac h t yp e are linearly indep en den t, s ince otherwise the corresp onding 2 × 2 minors formed by initial data m ust b e equal to zero. Therefore, basic vec tors of eac h type are lin early indep end en t. Giv en arbitrary generic initial d ata, we will d etermine the v alues of the scalar f ield u at the r emaining p oints of the elemen tary cub e u nder consideration according to relations (4) and mark the p oints at wh ic h the v alues of the f ield ha v e already b een found . W e will also shade eac h line in this elementa ry cub e if the v ector of v alues of the f ield u at the p oin ts of this lin e is a linear combinatio n of the vect ors of v alues of the f ield u at the points of the t w o basic lines of the same t ype (for the co ordinates of v ectors of v alues of the f ield u at the p oin ts of lines of the same t yp e, there is a natural ascending order of the resp ectiv e co ordin ate x , y or z ). First of all, in the elementa ry cub e un der consideration we must mark all lattice p oints at wh ic h the initial data are giv en and shade all basic lines of all thr ee t yp es b y the v ery def in ition of this pro cedure. It is ob vious that if carryin g out suc h a pr o cedure for generic in itial data yields all the lattice p oin ts marked and all the lines of all the typ es shaded in the elemen tary cub e un der consideration, then the theorem will b e pr o v ed, b ecause in this case, for any thr e e lines of the same typ e in this elementar y cub e (and, hence, for an y elemen tary 3 × 3 square in this elementa ry c ub e, b ent or unb en t), the determinan t of the matrix of v alues of the scalar f iel d u at the p oin ts of these lines will v anish, and this is even more than is required for the consistency of the corresp ond ing d iscrete equation. Th us, in this case, as a matter of fact, we will pro v e ev en a c onsider ably str onger principle of c onsistency on th e cubic lattic e Z 3 for determ inants and for the nonline ar discr ete e quation (4). It remains to shade all the lines of all the types in the element ary cub e un der consideration. F or this purp ose, it is necessary to consider consecutiv ely at least 13 elementa ry 3 × 3 s quares (b ent and un b en t) of our elementa ry cub e determinin g v alues of the f ield u a t 13 un mark ed p oin ts. Let us consider the el emen tary 3 × 3 square { ( i, 0 , 0) , ( i, 0 , 1) , ( i, 0 , 2) , i = 0 , 1 , 2 } in our cub e (a face of the cub e). In this elemen tary square the v alues of the f ield u are given at eight p oin ts and the v alue of the f ield u at the r emaining nin th p oin t (2 , 0 , 2) is determined b y relation (4), 8 O.I. Mokhov i.e., by the condition that the determinant of the matrix of v al ues of the f ield at the lattice p oints of this elemen tary 3 × 3 square v anishes, b ecause the corr esp ondin g 2 × 2 minor formed b y initial data is not equal to zero. S ince b asic vec tors of th e same type are lin early indep en- den t, the v ector of v alues of the f ield u at the p oin ts of the lin e { ( i, 0 , 2) , 0 ≤ i ≤ 2 } is a linear com bination of the v ectors of v alues of the f ie ld u at the p oint s of the tw o b asic lines of the same t yp e, { ( i, 0 , 0 ) , 0 ≤ i ≤ 2 } and { ( i, 0 , 1) , 0 ≤ i ≤ 2 } , situate d in the giv en eleme n tary 3 × 3 square; i.e., w e can mark th e p oin t (2 , 0 , 2) and shade the line { ( i, 0 , 2) , 0 ≤ i ≤ 2 } . Moreo v er, the obtained v ector of v alues of the f ield u at the p oin ts of the line { ( i, 0 , 2) , 0 ≤ i ≤ 2 } f orms a lin early in dep end en t pair of v ectors with eac h of the basic v ectors of the same t yp e, sin ce otherwise the corresp ond ing 2 × 2 minors formed by initial data m u st b e equal to zero. Similarly , since b asic ve ctors of the same t yp e are linearly indep en den t, it follo w s imme- diately from v anishing the determinant of the matrix of v alues of the f ield at the lattice p oints of this elemen tary 3 × 3 square that the v ecto r of v alues of the f ie ld u at the p oin ts of the line { (2 , 0 , k ) , 0 ≤ k ≤ 2 } is a linear co m bination of the v ectors of v alues of the f ield u at the p oin ts of th e other t w o lines of this elemen tary 3 × 3 squ are, namely , the t w o basic lines of the same t yp e, { (0 , 0 , k ) , 0 ≤ k ≤ 2 } and { (1 , 0 , k ) , 0 ≤ k ≤ 2 } , situated in the given element ary 3 × 3 square; i.e., we can shade the lin e { (2 , 0 , k ) , 0 ≤ k ≤ 2 } . Moreov er, the obtained v ecto r of v alues of the f ield u at th e p oin ts of the line { (2 , 0 , k ) , 0 ≤ k ≤ 2 } forms a linearly indep endent p air of v ectors with eac h of the basic vec tors of the same t yp e, since otherwise the co rresp onding 2 × 2 minors formed by initial data must b e equal to zero. Let us consider the b ent elemen tary 3 × 3 square { (1 , 0 , k ) , (0 , 0 , k ) , (0 , 1 , k ) , k = 0 , 1 , 2 } in our cub e. I n this elemen tary square the v alues of the f ield u are giv en at eigh t p oin ts and the v alue of the f ield u at the remaining ninth p oin t (0 , 1 , 2) is determined b y relation (4), i.e., b y the condition that the determinant of the matrix of v alues of the f ield at the lattice p oin ts of this b en t elementa ry 3 × 3 square v anishes, b ecause the corresp onding 2 × 2 minor formed b y initial data is not equal to zero. Since b asic v ecto rs of the same type are linearly ind ep end en t, the v ecto r of v alues of the f ield u at the p oints of the line { (0 , 1 , k ) , 0 ≤ k ≤ 2 } is a linear com bination of the v ecto rs of v alues of the f ield u at the p oin ts of the t w o basic lines of the s ame t yp e, { (0 , 0 , k ) , 0 ≤ k ≤ 2 } and { (1 , 0 , k ) , 0 ≤ k ≤ 2 } , s ituated in the giv en b ent elemen tary 3 × 3 square; i.e., w e can mark the p oin t (0 , 1 , 2) and shade the lin e { (0 , 1 , k ) , 0 ≤ k ≤ 2 } . Moreo v er, the obtained v ec tor of v a lues of the f ield u at the p oin ts of the line { (0 , 1 , k ) , 0 ≤ k ≤ 2 } forms a lin early in dep end en t pair of v ectors with eac h of the basic v ectors of the same t yp e, sin ce otherwise the corresp ond ing 2 × 2 minors formed by initial data m u st b e equal to zero. No w we consider another b ent elemen ta ry 3 × 3 squ are { (1 , 1 , k ) , (1 , 0 , k ) , (0 , 0 , k ) , k = 0 , 1 , 2 } in our cub e. In this elemen tary squ are the v alues of the f ield u are given at eigh t p oints and the v alue of the f ield u at the remaining nin th p oint (1 , 1 , 2) is determined b y relation (4), i.e., b y the cond ition that the determinant of the matrix of v alues of the f ield at the lattice p oints of this b ent elemen tary 3 × 3 square v anishes, b ecause the corresp ondin g 2 × 2 min or formed by initial data is not equal to zero. Since b asic v ecto rs of the same type are linearly ind ep end en t, the v ecto r of v alues of the f ield u at the p oints of the line { (1 , 1 , k ) , 0 ≤ k ≤ 2 } is a linear com bination of the v ecto rs of v alues of the f ield u at the p oin ts of the t w o basic lines of the s ame t yp e, { (1 , 0 , k ) , 0 ≤ k ≤ 2 } and { (0 , 0 , k ) , 0 ≤ k ≤ 2 } , s ituated in the giv en b ent elemen tary 3 × 3 square; i.e., w e can mark the p oin t (1 , 1 , 2) and shade the lin e { (1 , 1 , k ) , 0 ≤ k ≤ 2 } . Moreo v er, the obtained v ec tor of v a lues of the f ield u at the p oin ts of the line { (1 , 1 , k ) , 0 ≤ k ≤ 2 } forms a lin early in dep end en t pair of v ectors with eac h of the basic v ectors of the same t yp e, sin ce otherwise the corresp ond ing 2 × 2 minors formed by initial data m u st b e equal to zero. Let u s consider one more b en t elemen tary 3 × 3 square { ( i, 0 , 0) , ( i, 0 , 1) , ( i, 1 , 1) , i = 0 , 1 , 2 } in our cub e. In this elemen tary squ are the v alues of the f ield u are given at eigh t p oints and the v alue of the f ield u at the remaining nin th p oint (2 , 1 , 1) is determined b y relation (4), i.e., b y the cond ition that the determinant of the matrix of v alues of the f ield at the lattice p oints On Initial Data in th e Problem of Consistency on Cubic Latti ces for 3 × 3 Determinan ts 9 of this b ent elemen tary 3 × 3 square v anishes, b ecause the corresp ondin g 2 × 2 min or formed by initial data is not equal to zero. Since b asic v ecto rs of the same type are linearly ind ep end en t, the v ector of v alues of the f ield u at the p oin ts of the line { ( i, 1 , 1) , 0 ≤ i ≤ 2 } is a linear com- bination of the vecto rs of v alues of the f ield u at th e p oin ts of the t w o basic lines of the same t yp e, { ( i, 0 , 0) , 0 ≤ i ≤ 2 } and { ( i, 0 , 1) , 0 ≤ i ≤ 2 } , situated in the giv en b ent elemen tary 3 × 3 square; i.e., w e can mark th e p oin t (2 , 1 , 1) and shade the line { ( i, 1 , 1) , 0 ≤ i ≤ 2 } . Moreo v er, the obtained v ector of v alues of the f ield u at the p oin ts of the line { ( i, 1 , 1) , 0 ≤ i ≤ 2 } f orms a lin early in dep end en t pair of v ectors with eac h of the basic v ectors of the same t yp e, sin ce otherwise the corresp ond ing 2 × 2 minors formed by initial data m u st b e equal to zero. Let us consider the next b ent elemen tary 3 × 3 square { ( i, 0 , 1) , ( i, 0 , 0 ) , ( i, 1 , 0) , i = 0 , 1 , 2 } in our cub e. I n this elemen tary square the v alues of the f ield u are giv en at eigh t p oin ts and the v alue of the f ield u at the remaining ninth p oin t (2 , 1 , 0) is determined b y relation (4), i.e., b y the condition that the determinant of the matrix of v alues of the f ield at the lattice p oin ts of this b en t elementa ry 3 × 3 square v anishes, b ecause the corresp onding 2 × 2 minor formed b y initial data is not equal to zero. Since b asic v ecto rs of the same type are linearly ind ep end en t, the v ector of v alues of the f ield u at the p oin ts of the line { ( i, 1 , 0) , 0 ≤ i ≤ 2 } is a linear com- bination of the vecto rs of v alues of the f ield u at th e p oin ts of the t w o basic lines of the same t yp e, { ( i, 0 , 0) , 0 ≤ i ≤ 2 } and { ( i, 0 , 1) , 0 ≤ i ≤ 2 } , situated in the giv en b ent elemen tary 3 × 3 square; i.e., w e can mark th e p oin t (2 , 1 , 0) and shade the line { ( i, 1 , 0) , 0 ≤ i ≤ 2 } . Moreo v er, the obtained v ector of v alues of the f ield u at the p oin ts of the line { ( i, 1 , 0) , 0 ≤ i ≤ 2 } f orms a lin early in dep end en t pair of v ectors with eac h of the basic v ectors of the same t yp e, sin ce otherwise the corresp ond ing 2 × 2 minors formed by initial data m u st b e equal to zero. Let us co nsider one more b en t ele men tary 3 × 3 square { (1 , j , 0) , (0 , j, 0) , (0 , j, 1) , j = 0 , 1 , 2 } in our cub e. In this elemen tary squ are the v alues of the f ield u are given at eigh t p oints and the v alue of the f ield u at the remaining nin th p oint (1 , 2 , 0) is determined b y relation (4), i.e., b y the cond ition that the determinant of the matrix of v alues of the f ield at the lattice p oints of this b ent elemen tary 3 × 3 square v anishes, b ecause the corresp ondin g 2 × 2 min or formed by initial data is not equal to zero. Since b asic v ecto rs of the same type are linearly ind ep end en t, the v ector of v alues of the f ield u at the p oin ts of the line { (1 , j , 0) , 0 ≤ j ≤ 2 } is a linear com bination of the v ecto rs of v alues of the f ield u at the p oin ts of the t w o basic lines of the s ame t yp e, { (0 , j, 0) , 0 ≤ j ≤ 2 } and { (0 , j, 1) , 0 ≤ j ≤ 2 } , s ituated in the give n b en t elemen tary 3 × 3 square; i.e., w e can mark the p oin t (1 , 2 , 0) and shade the line { (1 , j, 0) , 0 ≤ j ≤ 2 } . Moreo v er, the obtained ve ctor of v alues of the f ield u at the p oin ts of the line { (1 , j, 0) , 0 ≤ j ≤ 2 } forms a lin early in dep end en t pair of v ectors with eac h of the basic v ectors of the same t yp e, sin ce otherwise the corresp ond ing 2 × 2 minors formed by initial data m u st b e equal to zero. Let u s consider the next b ent elemen tary 3 × 3 square { (0 , j, 0) , (0 , j , 1) , (1 , j, 1) , j = 0 , 1 , 2 } in our cub e. In this elemen tary squ are the v alues of the f ield u are given at eigh t p oints and the v alue of the f ield u at the remaining nin th p oint (1 , 2 , 1) is determined b y relation (4), i.e., b y the cond ition that the determinant of the matrix of v alues of the f ield at the lattice p oints of this b ent elemen tary 3 × 3 square v anishes, b ecause the corresp ondin g 2 × 2 min or formed by initial data is not equal to zero. Since b asic v ecto rs of the same type are linearly ind ep end en t, the v ector of v alues of the f ield u at the p oin ts of the line { (1 , j , 1) , 0 ≤ j ≤ 2 } is a linear com bination of the v ecto rs of v alues of the f ield u at the p oin ts of the t w o basic lines of the s ame t yp e, { (0 , j, 0) , 0 ≤ j ≤ 2 } and { (0 , j, 1) , 0 ≤ j ≤ 2 } , s ituated in the give n b en t elemen tary 3 × 3 square; i.e., w e can mark the p oin t (1 , 2 , 1) and shade the line { (1 , j, 1) , 0 ≤ j ≤ 2 } . Moreo v er, the obtained ve ctor of v alues of the f ield u at the p oin ts of the line { (1 , j, 1) , 0 ≤ j ≤ 2 } forms a lin early in dep end en t pair of v ectors with eac h of the basic v ectors of the same t yp e, sin ce otherwise the corresp ond ing 2 × 2 minors formed by initial data m u st b e equal to zero. Note that the ve ctors of v alues of the f ield u at the p oin ts of the shaded lines { (1 , j, 0) , 0 ≤ j ≤ 2 } and { (1 , j, 1) , 0 ≤ j ≤ 2 } , { ( i, 1 , 0) , 0 ≤ i ≤ 2 } and { ( i, 1 , 1) , 0 ≤ i ≤ 2 } , { (0 , 1 , k ) , 0 ≤ k ≤ 2 } 10 O.I. Mokhov and { (1 , 1 , k ) , 0 ≤ k ≤ 2 } , { (2 , 0 , k ) , 0 ≤ k ≤ 2 } and { (1 , 1 , k ) , 0 ≤ k ≤ 2 } , are linearly indep en- den t, s ince otherwise the corresp onding 2 × 2 minors formed by initial d ata m ust b e equal to zero. Let us consid er one more elemen tary 3 × 3 s quare { (0 , j, 0) , (0 , j, 1) , (0 , j, 2) , j = 0 , 1 , 2 } in our cub e (a face of the cub e). In this elemen ta ry square at the presen t momen t the v alues of the f ield u are already determined at eigh t p oin ts and the v alue of the f ield u at the remaining nin th p oin t (0 , 2 , 2) is determined b y relation (4), i.e., b y the condition that the determinant of the m atrix of v alues of the f ield at the lattice p oint s of this elementa ry 3 × 3 squ are v anishes, b ecause the corresp onding 2 × 2 m inor formed by initial data is not equal to zero. S ince basic v ectors of the same t yp e are linearly indep end en t, the v ector of v alues of the f ield u at the p oin ts of the line { (0 , j, 2) , 0 ≤ j ≤ 2 } is a linear co m bination of the v ec tors of v alues of the f ield u at the p oin ts of the tw o basic lines of the s ame t yp e, { (0 , j, 0) , 0 ≤ j ≤ 2 } and { (0 , j, 1) , 0 ≤ j ≤ 2 } , situated in the giv en elementa ry 3 × 3 squ are; i.e., w e can mark the p oin t (0 , 2 , 2) and shade the line { (0 , j, 2 ) , 0 ≤ j ≤ 2 } . But in this case w e do not state that the obtained v ector of v al ues of the f iel d u at the p oin ts of the line { (0 , j, 2) , 0 ≤ j ≤ 2 } form s linearly indep endent pairs of v ectors with basic v ecto rs of the same t yp e. W e note th at if the v ect or of v alues of the f ield u at the lattice p oint s of an arbitrary line is a linear com b ination of the v ecto rs of v alues of the f ield u at the lattic e p oint s of t w o sh aded lines of the same t yp e, then this v ector is a linear combinatio n of the v ecto rs of v alues of the f ield u at the p oin ts of the tw o basic lines of the same t yp e. Th is follo ws immediately from the fact that ea c h v ec tor of v alues of the f ield u at the p oint s of an arbitrary shaded lin e is a linear com bination of the v ectors of v alues of the f ie ld u at the p oint s of the tw o b asic lines of the same t yp e. Since the determinant of the matrix of v alues of th e f ield at the p oints of the elementa ry 3 × 3 s quare { (0 , j, 0) , (0 , j, 1) , (0 , j, 2) , j = 0 , 1 , 2 } (on a face of our cub e) v anishes and, as it w as noted a b o v e, the v ectors of v alues of the f ield u at th e p oints of the t w o sh aded lines, { (0 , 0 , k ) , 0 ≤ k ≤ 2 } and { (0 , 1 , k ) , 0 ≤ k ≤ 2 } , are linearly indep end en t, it follo w s immediately that the v ector of v alues of the f ield u at the p oin ts of the line { (0 , 2 , k ) , 0 ≤ k ≤ 2 } is a lin- ear combinatio n of th e v ectors of v alues of the f ield u at the p oints of these t w o lines of this elemen tary 3 × 3 square, namely , t w o shaded lines of the same t yp e, { (0 , 0 , k ) , 0 ≤ k ≤ 2 } and { (0 , 1 , k ) , 0 ≤ k ≤ 2 } , s ituated in the giv en elemen tary 3 × 3 square; i.e., w e can shade the line { (0 , 2 , k ) , 0 ≤ k ≤ 2 } . Moreo v er, the obtained v ecto r of v alues of the f ield u at the p oin ts of the line { (0 , 2 , k ) , 0 ≤ k ≤ 2 } forms a linearly indep endent pair of v ectors with the vecto r of v alues of the f iel d u at the p oints of the basic line { (0 , 0 , k ) , 0 ≤ k ≤ 2 } and also with the ve ctor of v alues of th e f ield u at the p oint s of the line { (0 , 1 , k ) , 0 ≤ k ≤ 2 } and the v ecto r of v alues of the f ield u at the p oin ts of the line { (1 , 1 , k ) , 0 ≤ k ≤ 2 } , sin ce otherwise the corresp onding 2 × 2 minors formed by initial data must b e equal to zero. In the elemen tary 3 × 3 square { ( i, 0 , 0) , ( i, 1 , 0) , ( i, 2 , 0) , i = 0 , 1 , 2 } of our cub e (on a f ace of the cub e) at the presen t moment the v alues of the f iel d u are already d etermined at eigh t p oin ts and the v alue of the f ield u at the r emaining nin th p oin t (2 , 2 , 0) is determined b y relation (4), i.e., by the condition that th e determinant of the m atrix of v alues of the f ield at the p oin ts of this elementa ry 3 × 3 square v anish es, b ecause th e co rresp onding 2 × 2 minor f ormed b y initial data is not equal to zero. Since, as it was n oted ab o v e, the v ectors of v alues of the f ield u at the p oin ts of the tw o shaded lines, { ( i, 0 , 0) , 0 ≤ i ≤ 2 } and { ( i, 1 , 0) , 0 ≤ i ≤ 2 } , are lin early indep en den t, the v ector of v alues of th e f ield u at the p oin ts of the line { ( i, 2 , 0 ) , 0 ≤ i ≤ 2 } is a linear com bination of the v ectors of v a lues of the f ield u at the p oin ts of these t w o shaded lines of the same t yp e, { ( i, 0 , 0) , 0 ≤ i ≤ 2 } and { ( i, 1 , 0) , 0 ≤ i ≤ 2 } , situated in the giv en elemen tary 3 × 3 square; i.e., we can mark the p oint (2 , 2 , 0) and shade the line { ( i, 2 , 0) , 0 ≤ i ≤ 2 } . But in this case we do not state th at the obtained vec tor of v alues of the f ield u at the p oints of the line { ( i, 2 , 0) , 0 ≤ i ≤ 2 } f orms linearly in dep end en t p airs of v ectors with other v ect ors o f v alues of the f ield u at the p oin ts of shaded lines of the same t yp e. On Initial Data in th e Problem of Consistency on Cubic Latti ces for 3 × 3 Determinan ts 11 Since the determinant of the matrix of v alues of th e f ield at the p oints of the elementa ry 3 × 3 square { ( i, 0 , 0) , ( i, 1 , 0 ) , ( i, 2 , 0) , i = 0 , 1 , 2 } (on a face of our cub e) v anish es and, a s it w as noted a b o v e, the v ectors of v alues of the f ield u at th e p oints of the t w o sh aded lines, { (0 , j, 0) , 0 ≤ j ≤ 2 } and { (1 , j, 0 ) , 0 ≤ j ≤ 2 } , are linearly indep endent, it follo ws immediately that the v ector of v alues of the f iel d u at the p oint s of the line { (2 , j, 0 ) , 0 ≤ j ≤ 2 } is a lin- ear com bination of the vect ors of v alues of the f iel d u at the p oint s of t w o other lines of this elemen tary 3 × 3 square, namely , tw o sh aded lines of the same t yp e, { (0 , j, 0) , 0 ≤ j ≤ 2 } and { (1 , j, 0) , 0 ≤ j ≤ 2 } , situated in the giv en elemen ta ry 3 × 3 square; i.e., w e can shade the line { (2 , j, 0) , 0 ≤ j ≤ 2 } . B ut in this case w e d o not sta te that the obtained vecto r of v alues of the f ield u at the points of the line { (2 , j, 0) , 0 ≤ j ≤ 2 } forms linearly ind ep endent pairs of ve ctors with ot her v ectors of v alues of th e f ield u at the p oin ts of shaded lines of the same t yp e. In the elemen ta ry 3 × 3 square { ( i, 0 , 1) , ( i, 1 , 1) , ( i, 2 , 1) , i = 0 , 1 , 2 } of our cub e (on a middle normal s ection of the cub e) at the pr esen t m omen t the v alues of the f ield u are already determined at eigh t p oin ts and the v alue of the f ield u at the remaining ninth p oint (2 , 2 , 1) is determined by relation (4), i.e., b y the condition th at the determinan t of the matrix of v alues of the f ield at the p oints of this elemen tary 3 × 3 square v anishes, b ecause the corresp onding 2 × 2 minor f ormed b y initial data is not equal to zero. Since, as it was noted ab o v e, the v ecto rs of v alues of th e f ield u at the p oint s of the t wo shaded lin es, { ( i, 0 , 1 ) , 0 ≤ i ≤ 2 } and { ( i, 1 , 1 ) , 0 ≤ i ≤ 2 } , are lin early indep en den t, the v ector of v alues of th e f ield u at the p oin ts of the line { ( i, 2 , 1 ) , 0 ≤ i ≤ 2 } is a linear com bination of the v ectors of v a lues of the f ield u at the p oin ts of these t w o shaded lines of the same t yp e, { ( i, 0 , 1) , 0 ≤ i ≤ 2 } and { ( i, 1 , 1) , 0 ≤ i ≤ 2 } , situated in the giv en elemen tary 3 × 3 squ are; i.e., we can mark the p oint (2 , 2 , 1) and sh ade the line { ( i, 2 , 1) , 0 ≤ i ≤ 2 } . Since the determinan t of th e matrix of v alues of the f ield at the p oint s of the elemen tary 3 × 3 square { ( i, 0 , 1) , ( i, 1 , 1 ) , ( i, 2 , 1) , i = 0 , 1 , 2 } (on a midd le normal section of our cub e) v anishes and, as it w as noted ab o v e, the v ectors of v alues of th e f ield u at the p oin ts of the t w o shaded lines, { (0 , j, 1) , 0 ≤ j ≤ 2 } and { (1 , j, 1 ) , 0 ≤ j ≤ 2 } , are linearly ind ep endent, it follo ws imme- diately that the v ecto r of v alues of the f ield u at the p oin ts of the line { (2 , j, 1) , 0 ≤ j ≤ 2 } is a linear com b ination of the ve ctors of v alues of the f ield u at the p oin ts of t w o other lines of this elemen tary 3 × 3 square, namely , tw o sh aded lines of the same t yp e, { (0 , j, 1) , 0 ≤ j ≤ 2 } and { (1 , j, 1) , 0 ≤ j ≤ 2 } , situated in the giv en elemen ta ry 3 × 3 square; i.e., w e can shade the line { (2 , j, 1) , 0 ≤ j ≤ 2 } . Let us consid er one more elemen tary 3 × 3 s quare { (1 , j, 0) , (1 , j, 1) , (1 , j, 2) , j = 0 , 1 , 2 } in our cub e (a midd le normal section of the cub e). In this elemen ta ry square at the presen t mo- men t the v al ues of th e f ield u are already determined at eight p oin ts and the v alue of th e f ield u at the remaining nint h p oint (1 , 2 , 2) is determined b y relation (4), i.e., b y the condi- tion that the determinan t of th e matrix of v alues of the f ield at the p oint s of this elemen ta ry 3 × 3 square v anish es, because the corresp onding 2 × 2 minor formed b y initial data is not equal to zero. S ince, as it w as noted ab o v e, the vecto rs of v alues of th e f iel d u at the p oin ts of the t w o shaded lines, { (1 , j, 0) , 0 ≤ j ≤ 2 } and { (1 , j, 1) , 0 ≤ j ≤ 2 } , are linea rly inde- p end en t, the v ector of v alues of the f ield u at the p oin ts of the lin e { (1 , j, 2) , 0 ≤ j ≤ 2 } is a linear com bination of the v ectors of v alues of the f ield u at the p oin ts of these t w o shaded lines of the same t y p e, { (1 , j, 0) , 0 ≤ j ≤ 2 } and { (1 , j, 1) , 0 ≤ j ≤ 2 } , situated in the gi v en elemen ta ry 3 × 3 square; i.e., we can mark the p oint (1 , 2 , 2) and shade the line { (1 , j, 2) , 0 ≤ j ≤ 2 } . Since the determinan t of th e matrix of v alues of the f ield at the p oint s of the elemen tary 3 × 3 square { (1 , j, 0) , (1 , j, 1) , (1 , j, 2) , j = 0 , 1 , 2 } (on a m iddle norm al section of our cub e) v anishes and, as it w as noted ab o v e, the v ectors of v alues of th e f ield u at the p oin ts of the t w o shaded lines, { (1 , 0 , k ) , 0 ≤ k ≤ 2 } and { (1 , 1 , k ) , 0 ≤ k ≤ 2 } , are linearly ind ep endent, it follo ws imme- diately that the v ecto r of v alues of the f iel d u at the p oin ts of the line { (1 , 2 , k ) , 0 ≤ k ≤ 2 } is a linear com bination of the vect ors of v alues of the f ield u at the p oints of t w o other lines of this 12 O.I. Mokhov elemen tary 3 × 3 square, namely , t w o shaded lines of the same t yp e, { (1 , 0 , k ) , 0 ≤ k ≤ 2 } and { (1 , 1 , k ) , 0 ≤ k ≤ 2 } , s ituated in the giv en elemen tary 3 × 3 square; i.e., w e can shade the line { (1 , 2 , k ) , 0 ≤ k ≤ 2 } . Let us consider one more elemen tary 3 × 3 squ are { ( i, 1 , 0) , ( i , 1 , 1) , ( i, 1 , 2) , i = 0 , 1 , 2 } in our cub e (a midd le normal section of the cub e). In this elemen ta ry square at the presen t mo- men t the v al ues of th e f ield u are already determined at eight p oin ts and the v alue of th e f ield u at the remaining nint h p oint (2 , 1 , 2) is determined b y relation (4), i.e., b y the condi- tion that the determinan t of th e matrix of v alues of the f ield at the p oint s of this elemen ta ry 3 × 3 square v anish es, because the corresp onding 2 × 2 minor formed b y initial data is not equal to zero. S ince, as it w as noted ab o v e, the vecto rs of v alues of th e f iel d u at the p oin ts of the t w o shaded lines, { ( i, 1 , 0) , 0 ≤ i ≤ 2 } and { ( i, 1 , 1) , 0 ≤ i ≤ 2 } , are linearly inde- p end en t, the v ector of v alues of the f ield u at the p oin ts of t he line { ( i, 1 , 2) , 0 ≤ i ≤ 2 } is a linear com bination of the v ectors of v alues of the f ield u at the p oin ts of these t w o shaded lines of the same t yp e, { ( i, 1 , 0) , 0 ≤ i ≤ 2 } and { ( i, 1 , 1) , 0 ≤ i ≤ 2 } , situated in the gi v en elemen ta ry 3 × 3 square; i.e., we can mark the p oint (2 , 1 , 2) and shade the line { ( i, 1 , 2) , 0 ≤ i ≤ 2 } . Since the determinan t of th e matrix of v alues of the f ield at the p oint s of the elemen tary 3 × 3 square { ( i, 1 , 0) , ( i, 1 , 1 ) , ( i, 1 , 2) , i = 0 , 1 , 2 } (on a midd le normal section of our cub e) v anishes and, as it w as noted ab o v e, the v ectors of v alues of th e f ield u at the p oin ts of the t w o shaded lines, { (0 , 1 , k ) , 0 ≤ k ≤ 2 } and { (1 , 1 , k ) , 0 ≤ k ≤ 2 } , are linearly ind ep endent, it follo ws imme- diately that the v ecto r of v alues of the f iel d u at the p oin ts of the line { (2 , 1 , k ) , 0 ≤ k ≤ 2 } is a linear com bination of the vect ors of v alues of the f ield u at the p oints of t w o other lines of this elemen tary 3 × 3 square, namely , t w o shaded lines of the same t yp e, { (0 , 1 , k ) , 0 ≤ k ≤ 2 } and { (1 , 1 , k ) , 0 ≤ k ≤ 2 } , s ituated in the giv en elemen tary 3 × 3 square; i.e., w e can shade the line { (2 , 1 , k ) , 0 ≤ k ≤ 2 } . It r emains to determine the v alue of the f ield u only at one p oint (2 , 2 , 2) of our cub e, and only three edges of th e cub e that con tain this p oint are s till u nshaded for the presen t. In order to d etermine the v alue of th e f ield u at th e remaining lattice p oint (2 , 2 , 2), it is necessary to sho w that in our cub e there is at least one elemen ta ry 3 × 3 square (b ent or unb en t) con taining this p oin t (2 , 2 , 2) and suc h that the corresp ond ing 2 × 2 m inor in the 3 × 3 matrix of v alues of the f ield u at the p oin ts of this elemen tary 3 × 3 square is not equ al to zero, since otherwise any v alue of the f ield u at the p oin t (2 , 2 , 2) could not b e determined. Note that in our cub e the lattic e p oint (2 , 2 , 2) is the only on e for whic h in eac h elementa ry 3 × 3 square (b ent or unb en t) conta ining this lattice p oin t the corresp ondin g 2 × 2 minor is not completely formed b y in itial data. Let u s pro v e that th e determinant of the 2 × 2 matrix of v alues of the f ield u at the p oint s { (0 , 1 , 1) , (1 , 1 , 1) , (0 , 2 , 1) , (1 , 2 , 1) } of our cub e is not equal to zero. Let us assume that it v a- nishes. In th is case, th e 2-v ec tors ( u 011 , u 021 ) and ( u 111 , u 121 ) must be linearly dep endent, and since the vec tor ( u 011 , u 021 ) of initial data must not b e zero (otherwise the corresp onding 2 × 2 minor formed by initial data m ust v anish), the v ecto r ( u 111 , u 121 ) is prop ortional to the ve ctor ( u 011 , u 021 ): ( u 111 , u 121 ) = λ ( u 011 , u 021 ) . (5) On the other hand , as it w as noted abov e, the vect or ( u 101 , u 111 , u 121 ) of v alues of the f ield u is a linear com bination of the basic v ecto rs of the s ame t yp e ( u 000 , u 010 , u 020 ) and ( u 001 , u 011 , u 021 ): ( u 101 , u 111 , u 121 ) = α ( u 000 , u 010 , u 020 ) + β ( u 001 , u 011 , u 021 ) , (6) and in this case ( u 111 , u 121 ) = α ( u 010 , u 020 ) + β ( u 011 , u 021 ) . On Initial Data in th e Problem of Consistency on Cubic Latti ces for 3 × 3 Determinan ts 13 Using relatio n (5) w e obtain λ ( u 011 , u 021 ) = α ( u 010 , u 020 ) + β ( u 011 , u 021 ) . Hence, since th e 2-v ec tors ( u 011 , u 021 ) and ( u 010 , u 020 ) are linearly indep endent (otherwise the corresp ondin g 2 × 2 minor formed by initial d ata must v anish ), we ha v e α = 0, i.e., relation (6) assumes the form ( u 101 , u 111 , u 121 ) = β ( u 001 , u 011 , u 021 ) , but this is imp ossible, b ecause in this case the corresp onding 2 × 2 m inor form ed b y initial data m ust v anish. Th us, it is pro v ed that the determinant of the 2 × 2 matrix of v alues of the f ield u at the p oints { (0 , 1 , 1) , (1 , 1 , 1) , (0 , 2 , 1) , (1 , 2 , 1) } of our cub e is not equal to zero. Let us co nsider the b en t elemen tary 3 × 3 square { ( i, 1 , 1) , ( i, 2 , 1) , ( i , 2 , 2) , i = 0 , 1 , 2 } in our cub e. In th is elemen tary squ are at the present momen t the v alues of the f iel d u are already determined at eigh t p oin ts and the v alue of th e f ield u at the remaining n in th p oin t (2 , 2 , 2) is determined b y relation (4), i.e., b y th e condition that the d eterminan t of the matrix of v alues of th e f ield at th e lattice p oints of th is elementa ry 3 × 3 square v anishes, b ecause w e p ro v ed that the corresp onding 2 × 2 minor is not equal to zero. Since the v ectors of v alues of the f ield u at the p oin ts of the t w o shad ed lines, { ( i, 2 , 1) , 0 ≤ i ≤ 2 } and { ( i, 1 , 1) , 0 ≤ i ≤ 2 } , are lin early indep endent (otherwise the nonzero 2 × 2 minor considered ab o v e m ust v anish), the vect or of v alues of the f ield u at the p oin ts of the li ne { ( i, 2 , 2) , 0 ≤ i ≤ 2 } is a linear com bination of the vecto rs of v alues of the f ield u at the p oint s of these tw o shaded lines of the same t yp e, { ( i, 2 , 1) , 0 ≤ i ≤ 2 } and { ( i, 1 , 1) , 0 ≤ i ≤ 2 } , situated in the g iv en b ent elemen tary 3 × 3 square; i.e., we can mark the p oin t (2 , 2 , 2) and s hade the line { ( i, 2 , 2) , 0 ≤ i ≤ 2 } . No w the v alues of the f ield u are determined already at all p oints of our cub e, and it r emains to s hade t wo ed ges of the cub e. A t f irst we prov e that the d eterminan t of the 2 × 2 matrix of v alues of the f ield u at the p oints { (0 , 1 , 1) , (1 , 1 , 1) , (0 , 1 , 2) , (1 , 1 , 2) } of our cub e is n ot equ al to zero. Let us assume that it v anishes. In this case, the 2-v ectors ( u 011 , u 012 ) and ( u 111 , u 112 ) m ust b e linearly de- p end en t. Moreo v er, both th ese v ectors are nonzero, since otherwise the corresp onding 2 × 2 minor formed b y initial data m ust v anish. Indeed, let us assume, for example, that the 2- v ector ( u 011 , u 012 ) is zero, i.e., u 011 = 0 , u 012 = 0 . T hen, in the b ent elementa ry 3 × 3 s quare { (0 , 1 , k ) , (0 , 0 , k ) , (1 , 0 , k ) , k = 0 , 1 , 2 } in our cub e, the determinan t of the matrix of v alues of the f ield u at the points of this b ent elemen ta ry 3 × 3 square i s e qual, except for sig n, to the p ro du ct of the v alue u 010 of th e f ie ld u by the corresp onding 2 × 2 min or formed by ini- tial data, and th is 2 × 2 minor is not equal to zero b y our cond ition on in itial data. Thus, as the d eterminan t of the m atrix of v alues of the f ield at the p oin ts of this b en t elemen- tary 3 × 3 s quare v anishes, it follo ws that u 010 = 0, a nd since u 011 = 0, the corresp onding 2 × 2 minor f ormed by initial data is equal to zero, b ut this is imp ossible. Sim ilarly , it can b e pro v ed that the 2 -v ector ( u 111 , u 112 ) is also nonzero. Indeed, let us assume that the 2- v ector ( u 111 , u 112 ) is zero, i.e., u 111 = 0 , u 112 = 0 . T hen, in the b ent elementa ry 3 × 3 s quare { (1 , 1 , k ) , (1 , 0 , k ) , (0 , 0 , k ) , k = 0 , 1 , 2 } in our cub e, the determinan t of the matrix of v alues of the f ield u at the p oints of this b ent elemen tary 3 × 3 s quare is equal, except f or sign, to the pro du ct of the v alue u 110 of the f ield u b y the corresp ond ing 2 × 2 min or formed by initial data, and this 2 × 2 minor is n ot equal to zero by our condition on initial data. Thus, as th e determinan t of the matrix of v alues of the f ield at the p oin ts of this b en t elementa ry 3 × 3 square v anishes, it follo ws that u 110 = 0, and since u 111 = 0, the corresp onding 2 × 2 m inor formed by initial d ata is equal to z ero, but this is imp ossible. So we hav e pro v ed that the 14 O.I. Mokhov 2-v ect ors ( u 011 , u 012 ) and ( u 111 , u 112 ) are nonzero. Moreo v er, these 2-v ectors m ust b e linearly dep end en t under our a ssumption. Therefore, eac h of t hese 2-v ectors is prop ortional t o the other: ( u 011 , u 012 ) = λ ( u 111 , u 112 ) , ( u 111 , u 112 ) = µ ( u 011 , u 012 ) . (7) On th e other h and, since the determinant of the matrix of v alues of the f ield at the p oint s of the b en t elemen tary 3 × 3 square { (1 , 0 , k ) , (1 , 1 , k ) , (0 , 1 , k ) , k = 0 , 1 , 2 } v anishes and, as it wa s noted ab o v e, th e ve ctors of v alues of the f ield u at the p oints of the t w o shaded lines, { (1 , 1 , k ) , 0 ≤ k ≤ 2 } and { (1 , 0 , k ) , 0 ≤ k ≤ 2 } , are linearly i ndep endent, it follo ws immediately that the ve ctor o f v alues of the f ield u at the p oin ts of the line { (0 , 1 , k ) , 0 ≤ k ≤ 2 } is a linear com b ination o f the vecto rs of v alues of the f ield u at the p oints of t wo other lines of this b en t elemen tary 3 × 3 squ are, n amely , tw o shaded lines of the same typ e, { (1 , 1 , k ) , 0 ≤ k ≤ 2 } and { (1 , 0 , k ) , 0 ≤ k ≤ 2 } , situated in the giv en b ent elemen tary 3 × 3 square: ( u 010 , u 011 , u 012 ) = α ( u 110 , u 111 , u 112 ) + β ( u 100 , u 101 , u 102 ) , (8) and in this case ( u 011 , u 012 ) = α ( u 111 , u 112 ) + β ( u 101 , u 102 ) . Using relatio n (7), w e obtain λ ( u 111 , u 112 ) = α ( u 111 , u 112 ) + β ( u 101 , u 102 ) . If the 2-v ectors ( u 111 , u 112 ) and ( u 101 , u 102 ) are lin early ind ep endent, i.e., the determinant of the matrix of v alues of the f ield u at the p oints { ( 1 , 1 , 1) , (1 , 0 , 1) , (1 , 0 , 2) , (1 , 1 , 2) } of our cub e is not equal to zero, then β = 0 and relation (8) assu mes the form ( u 010 , u 011 , u 012 ) = α ( u 110 , u 111 , u 112 ) , but this is imp ossible, since in this case the corresp onding 2 × 2 minor formed b y in itial data m ust b e equal to zero. Hence, under our assumptions the 2-ve ctors ( u 111 , u 112 ) and ( u 101 , u 102 ) m ust b e lin early dep endent , i.e., the determinan t of 2 × 2 matrix of v alues of the f ield u at the p oints { (1 , 1 , 1) , (1 , 0 , 1) , (1 , 0 , 2) , ( 1 , 1 , 2) } of our cub e v anishes. Since the 2-vec tor ( u 111 , u 112 ) is nonzero and the d eterminan t of the 2 × 2 matrix of v alues of the f ield u at the p oin ts { (1 , 1 , 1) , (1 , 0 , 1) , (1 , 0 , 2) , (1 , 1 , 2) } of our cub e v an ishes, it follo w s that the 2-v ector ( u 101 , u 102 ) is prop ortional to the 2-ve ctor ( u 111 , u 112 ): ( u 101 , u 102 ) = ν ( u 111 , u 112 ) . (9) Since the determinan t of the matrix of v alues of the f ie ld at the p oin ts of the b en t elementa ry 3 × 3 square { (1 , 1 , k ) , (0 , 1 , k ) , (0 , 0 , k ) , k = 0 , 1 , 2 } v anish es and, a s it was n oted ab ov e, the v ectors of v alues of the f iel d u at the p oin ts of the t w o shaded lines, { (0 , 1 , k ) , 0 ≤ k ≤ 2 } and { (0 , 0 , k ) , 0 ≤ k ≤ 2 } , are linearly indep end en t, it f ollo w s immediately that the v ector of v alues of the f ield u at the p oin ts of the line { (1 , 1 , k ) , 0 ≤ k ≤ 2 } is a linear com bination of th e ve ctors of v alues of the f ield u at th e p oint s of tw o other lines of this b ent elemen tary 3 × 3 square, namely , tw o shaded lines of the s ame typ e, { (0 , 1 , k ) , 0 ≤ k ≤ 2 } and { (0 , 0 , k ) , 0 ≤ k ≤ 2 } , situated in the giv en b ent elemen tary 3 × 3 square: ( u 110 , u 111 , u 112 ) = α ( u 010 , u 011 , u 012 ) + β ( u 000 , u 001 , u 002 ) , (10) On Initial Data in th e Problem of Consistency on Cubic Latti ces for 3 × 3 Determinan ts 15 and in this case ( u 111 , u 112 ) = α ( u 011 , u 012 ) + β ( u 001 , u 002 ) . Using relatio n (7), w e obtain µ ( u 011 , u 012 ) = α ( u 011 , u 012 ) + β ( u 001 , u 002 ) . If the 2-v ectors ( u 011 , u 012 ) and ( u 001 , u 002 ) are lin early ind ep endent, i.e., the determinant of the matrix of v alues of the f ield u at the p oints { ( 0 , 1 , 1) , (0 , 0 , 1) , (0 , 0 , 2) , (0 , 1 , 2) } of our cub e is not equal to zero, then β = 0 and relation (10) assum es the form ( u 110 , u 111 , u 112 ) = α ( u 010 , u 011 , u 012 ) , but this is imp ossible, since in this case the corresp onding 2 × 2 minor formed b y in itial data m ust b e equal to zero. Hence, under our assumptions the 2-ve ctors ( u 011 , u 012 ) and ( u 001 , u 002 ) m ust b e linearly dep end en t, i.e., the determinant of the 2 × 2 matrix of v alues of the f ield u at the p oints { (0 , 1 , 1) , (0 , 0 , 1) , (0 , 0 , 2) , ( 0 , 1 , 2) } of our cub e v anishes. Since the 2-vec tor ( u 011 , u 012 ) is nonzero and the d eterminan t of the 2 × 2 matrix of v alues of the f ield u at the p oin ts { (0 , 1 , 1) , (0 , 0 , 1) , (0 , 0 , 2) , (0 , 1 , 2) } of our cub e v an ishes, it follo w s that the 2-v ector ( u 001 , u 002 ) is prop ortional to the 2-ve ctor ( u 011 , u 012 ): ( u 001 , u 002 ) = κ ( u 011 , u 012 ) . Using relatio n (7), w e obtain ( u 001 , u 002 ) = κ ( u 011 , u 012 ) = κ λ ( u 111 , u 112 ) , and from (9) w e hav e ( u 101 , u 102 ) = ν ( u 111 , u 112 ) , i.e., the 2-v ectors ( u 001 , u 002 ) and ( u 101 , u 102 ) are linearly dep end en t, but this is impossib le, since in this case the corresp onding 2 × 2 m inor f ormed by initial d ata must b e equal to zero. Th us, it is pro v ed that the determinant of the 2 × 2 matrix of v alues of the f ield u at the p oints { (0 , 1 , 1) , (1 , 1 , 1) , (0 , 1 , 2) , (1 , 1 , 2) } of our cub e is not equal to zero. Since the determinan t of the matrix of v alues of the f ield u at the p oints of th e b ent element ary 3 × 3 squ are { ( i, 0 , 2) , ( i, 1 , 2) , ( i, 1 , 1) , i = 0 , 1 , 2 } v anishes (the three lines { ( i, 0 , 2) , i = 0 , 1 , 2 } , { ( i, 1 , 2) , i = 0 , 1 , 2 } and { ( i, 1 , 1) , i = 0 , 1 , 2 } are shaded in this b ent elemen tary 3 × 3 square), it follo w s immediately that the three v ectors of v alues of the f ield u at the p oint s of the b en t lines { (0 , 0 , 2) , (0 , 1 , 2) , (0 , 1 , 1) } , { (1 , 0 , 2) , (1 , 1 , 2) , (1 , 1 , 1) } and { (2 , 0 , 2) , (2 , 1 , 2) , (2 , 1 , 1) } are linearly dep endent; moreo v er, the v ectors of v alues o f the f ield u at th e p oin ts of the b en t lines { (0 , 0 , 2) , (0 , 1 , 2) , (0 , 1 , 1) } and { (1 , 0 , 2) , (1 , 1 , 2) , (1 , 1 , 1) } are linearly indep endent , since otherwise the nonzero determinant of the 2 × 2 matrix of v alues of the f ield u at the p oint s { (0 , 1 , 1) , (1 , 1 , 1) , (0 , 1 , 2) , (1 , 1 , 2) } of our cub e m ust v anish. Thus, the vect or of v alues of the f ield u at the p oints of the b ent line { (2 , 0 , 2) , (2 , 1 , 2) , (2 , 1 , 1) } is a linear combination of the v ectors of v alues of the f ield u at the p oints of the b ent lines { (0 , 0 , 2) , (0 , 1 , 2) , (0 , 1 , 1) } and { (1 , 0 , 2) , (1 , 1 , 2) , (1 , 1 , 1) } : ( u 202 , u 212 , u 211 ) = α ( u 002 , u 012 , u 011 ) + β ( u 102 , u 112 , u 111 ) . (11) Similarly , since the determinan t of the matrix of v a lues of the f ield at the p oin ts of the b en t elemen tary 3 × 3 square { ( i, 2 , 2) , ( i, 1 , 2) , ( i, 1 , 1) , i = 0 , 1 , 2 } v anishes (the three lines { ( i, 2 , 2) , i = 16 O.I. Mokhov 0 , 1 , 2 } , { ( i, 1 , 2) , i = 0 , 1 , 2 } and { ( i, 1 , 1) , i = 0 , 1 , 2 } are s haded in this b ent elemen tary 3 × 3 square), it follo ws immediately that the vec tors of v alues of the f ie ld u at the p oin ts of the b ent lines { (0 , 2 , 2) , (0 , 1 , 2) , (0 , 1 , 1 ) } , { (1 , 2 , 2) , (1 , 1 , 2) , ( 1 , 1 , 1) } and { ( 2 , 2 , 2) , (2 , 1 , 2 ) , (2 , 1 , 1) } are linearly dep en den t; moreov er, the vect ors of v alues of the f ield u at the p oints of the b ent lines { (0 , 2 , 2) , (0 , 1 , 2) , (0 , 1 , 1) } and { (1 , 2 , 2) , (1 , 1 , 2) , (1 , 1 , 1) } are linearly indep endent , since otherwise the nonzero determinant of the 2 × 2 matrix of v alues of the f ield u at the p oint s { (0 , 1 , 1) , (1 , 1 , 1) , (0 , 1 , 2) , (1 , 1 , 2) } of our cub e m ust v anish. Thus, in the b ent elemen tary 3 × 3 square u nder consider ation, the v ector of v alues of th e f ield u at the p oints of the b ent line { (2 , 2 , 2) , (2 , 1 , 2) , (2 , 1 , 1) } is a linear com bination of th e v ectors of v alues of the f ield u at th e p oints of the b ent lines { (0 , 2 , 2) , (0 , 1 , 2) , (0 , 1 , 1 ) } and { (1 , 2 , 2) , (1 , 1 , 2) , (1 , 1 , 1) } : ( u 222 , u 212 , u 211 ) = γ ( u 022 , u 012 , u 011 ) + δ ( u 122 , u 112 , u 111 ) . (12) F rom relations (11 ) and (12), w e obtain resp ectiv ely ( u 212 , u 211 ) = α ( u 012 , u 011 ) + β ( u 112 , u 111 ) and ( u 212 , u 211 ) = γ ( u 012 , u 011 ) + δ ( u 112 , u 111 ) , whence it follo ws immediately that α = γ and β = δ , since the determinant of the 2 × 2 matrix of v alues of the f ield u at the p oin ts { (0 , 1 , 1) , (1 , 1 , 1) , (0 , 1 , 2) , (1 , 1 , 2) } of our cub e is not equal to zero and the 2-v ec tors ( u 011 , u 012 ) an d ( u 111 , u 112 ) are linearly indep end en t. F rom relations (11) and (12), we obtain resp ectiv ely ( u 202 , u 212 ) = α ( u 002 , u 012 ) + β ( u 102 , u 112 ) (13) and ( u 222 , u 212 ) = γ ( u 022 , u 012 ) + δ ( u 122 , u 112 ) . (14) Since α = γ and β = δ , from r elations (13) and (14) ( u 202 , u 212 , u 222 ) = α ( u 002 , u 012 , u 022 ) + β ( u 102 , u 112 , u 122 ) , i.e., the v ecto r of v alues of the f ield u at the p oin ts of the line { (2 , 2 , 2) , (2 , 1 , 2 ) , ( 2 , 0 , 2) } is a linear com b ination of the v ectors of v alues of the f ield u at the p oin ts of the t w o sh aded lines { (0 , 2 , 2) , (0 , 1 , 2) , (0 , 0 , 2) } and { (1 , 2 , 2) , (1 , 1 , 2) , (1 , 0 , 2) } , and hence we can sh ade also the line { (2 , 2 , 2) , (2 , 1 , 2) , (2 , 0 , 2) } in our cub e. Let us p ro v e n o w that the d eterminan t of the 2 × 2 m atrix of v alues of the f ield u at th e p oints { (1 , 0 , 1) , (1 , 1 , 1) , (2 , 1 , 1) , ( 2 , 0 , 1) } of our cub e is not equal to zero. Let us assume that it v anish es. In this case, the 2-v ectors ( u 111 , u 211 ) and ( u 101 , u 201 ) must be linearly dep endent. Moreo v er, th e vec tor ( u 101 , u 201 ) is nonzero, sin ce otherwise the corresp onding 2 × 2 minor form ed b y initial data m ust v a nish. In this case the 2-v ector ( u 111 , u 211 ) must b e pr op ortional to the 2-v ect or ( u 101 , u 201 ): ( u 111 , u 211 ) = λ ( u 101 , u 201 ) . (15) On th e other h and, since the determinant of the matrix of v alues of the f ield at the p oint s of th e b ent ele men tary 3 × 3 square { ( i, 0 , 0) , ( i, 0 , 1) , ( i, 1 , 1) , i = 0 , 1 , 2 } v anishes and th e vec- tors of v alues of the f ield u at th e points o f the t wo basic l ines { ( i, 0 , 0) , 0 ≤ i ≤ 2 } and { ( i, 0 , 1) , 0 ≤ i ≤ 2 } are linearly ind ep endent, it follo ws immediate ly th at the v ector of v alues of the f iel d u at the p oin ts of the line { ( i, 1 , 1) , 0 ≤ i ≤ 2 } is a linear com b ination of th e vect ors o f On Initial Data in th e Problem of Consistency on Cubic Latti ces for 3 × 3 Determinan ts 17 v alues of the f ield u at the p oint s of the t w o lines of this b ent elemen tary 3 × 3 square, namely , the tw o b asic lines of the same typ e { ( i, 0 , 0) , 0 ≤ i ≤ 2 } and { ( i, 0 , 1) , 0 ≤ i ≤ 2 } , situated in the giv en b en t elemen tary 3 × 3 sq uare: ( u 011 , u 111 , u 211 ) = α ( u 001 , u 101 , u 201 ) + β ( u 000 , u 100 , u 200 ) , (16) and in this case ( u 111 , u 211 ) = α ( u 101 , u 201 ) + β ( u 100 , u 200 ) . Using relatio n (15), we obtain λ ( u 101 , u 201 ) = α ( u 101 , u 201 ) + β ( u 100 , u 200 ) . Since the 2-v ect ors ( u 101 , u 201 ) and ( u 100 , u 200 ) are linearly in dep end en t (otherwise the corre- sp ond ing 2 × 2 m inor formed by initial data must v anish), we hav e β = 0 and relation (16) assumes the form ( u 011 , u 111 , u 211 ) = α ( u 001 , u 101 , u 201 ) , but this is imp ossible, b ecause in this case the corresp onding 2 × 2 m inor form ed b y initial data m ust v anish. Th us, it is pro v ed that the determinant of the 2 × 2 matrix of v alues of the f ield u at the p oints { (1 , 0 , 1) , (1 , 1 , 1) , (2 , 1 , 1) , (2 , 0 , 1) } of our cub e is not equal to zero. Since the determinan t of the matrix of v alues of the f ie ld at the p oin ts of the b en t elementa ry 3 × 3 s quare { (2 , j, 0) , (2 , j, 1) , (1 , j , 1 ) , j = 0 , 1 , 2 } v anishes (t he three lines { ( 2 , j, 0) , j = 0 , 1 , 2 } , { (2 , j, 1) , j = 0 , 1 , 2 } and { ( 1 , j, 1) , j = 0 , 1 , 2 } are shaded in this b en t element ary 3 × 3 square), it f ollo ws immediately that the v ectors of v alues of the f ield u at the p oints of the b en t lines { (2 , 0 , 0) , (2 , 0 , 1) , (1 , 0 , 1) } , { (2 , 1 , 0) , (2 , 1 , 1) , (1 , 1 , 1) } and { (2 , 2 , 0) , (2 , 2 , 1) , (1 , 2 , 1) } are linearly dep endent; moreo v er, the v ectors of v alues o f the f ield u at th e p oin ts of the b en t lines { (2 , 0 , 0) , (2 , 0 , 1) , (1 , 0 , 1) } and { (2 , 1 , 0) , (2 , 1 , 1) , (1 , 1 , 1) } are linearly indep endent , since otherwise the nonzero determinant of the 2 × 2 matrix of v alues of the f ield u at the p oint s { (1 , 0 , 1) , (1 , 1 , 1) , (2 , 1 , 1) , (2 , 0 , 1) } of our cub e m ust v anish. Thus, the vect or of v alues of the f ield u at the p oints of the b ent line { (2 , 2 , 0) , (2 , 2 , 1) , (1 , 2 , 1) } is a linear combination of the v ectors of v alues of the f ield u at the p oints of the b ent lines { (2 , 0 , 0) , (2 , 0 , 1) , (1 , 0 , 1) } and { (2 , 1 , 0) , (2 , 1 , 1) , (1 , 1 , 1) } : ( u 220 , u 221 , u 121 ) = α ( u 200 , u 201 , u 101 ) + β ( u 210 , u 211 , u 111 ) . (17) Similarly , since the determinan t of the matrix of v alues of the f ie ld at the p oint s of the b ent el- emen tary 3 × 3 s quare { (2 , j, 2) , (2 , j, 1) , (1 , j, 1) , j = 0 , 1 , 2 } v anishes (the thr ee lines { (2 , j, 2) , j = 0 , 1 , 2 } , { (2 , j, 1) , j = 0 , 1 , 2 } and { (1 , j, 1) , j = 0 , 1 , 2 } are shaded in this b ent element ary 3 × 3 square), it follo ws immediately that the vec tors of v alues of the f ie ld u at the p oin ts of the b ent lines { (1 , 0 , 1) , (2 , 0 , 1) , (2 , 0 , 2 ) } , { (1 , 1 , 1) , (2 , 1 , 1) , ( 2 , 1 , 2) } and { ( 1 , 2 , 1) , (2 , 2 , 1 ) , (2 , 2 , 2) } are linearly dep en den t; moreov er, the vect ors of v alues of the f ield u at the p oints of the b ent lines { (1 , 0 , 1) , (2 , 0 , 1) , (2 , 0 , 2) } and { (1 , 1 , 1) , (2 , 1 , 1) , (2 , 1 , 2) } are linearly indep endent , since otherwise the nonzero determinant of the 2 × 2 matrix of v alues of the f ield u at the p oint s { (1 , 0 , 1) , (1 , 1 , 1) , (2 , 1 , 1) , (2 , 0 , 1) } of our cub e m ust v anish. Thus, in the b ent elemen tary 3 × 3 square u nder consider ation, the v ector of v alues of th e f ield u at the p oints of the b ent line { (1 , 2 , 1) , (2 , 2 , 1) , (2 , 2 , 2) } is a linear com bination of th e v ectors of v alues of the f ield u at th e p oints of the b ent lines { (1 , 0 , 1) , (2 , 0 , 1) , (2 , 0 , 2 ) } and { (1 , 1 , 1) , (2 , 1 , 1) , (2 , 1 , 2) } : ( u 222 , u 221 , u 121 ) = γ ( u 202 , u 201 , u 101 ) + δ ( u 212 , u 211 , u 111 ) . (18) 18 O.I. Mokhov F rom relations (17 ) and (18), w e obtain resp ectiv ely ( u 221 , u 121 ) = α ( u 201 , u 101 ) + β ( u 211 , u 111 ) and ( u 221 , u 121 ) = γ ( u 201 , u 101 ) + δ ( u 211 , u 111 ) , whence it follo ws immediately that α = γ and β = δ , since the determinant of the 2 × 2 matrix of v alues of the f ield u at the p oin ts { (1 , 0 , 1) , (1 , 1 , 1) , (2 , 1 , 1) , (2 , 0 , 1) } of our cub e is not equal to zero and the 2-v ec tors ( u 201 , u 101 ) an d ( u 211 , u 111 ) are linearly indep end en t. F rom relations (17) and (18), we obtain resp ectiv ely ( u 220 , u 221 ) = α ( u 200 , u 201 ) + β ( u 210 , u 211 ) (19) and ( u 222 , u 221 ) = γ ( u 202 , u 201 ) + δ ( u 212 , u 211 ) . (20) Since α = γ and β = δ , from r elations (19) and (20) ( u 222 , u 221 , u 220 ) = α ( u 202 , u 201 , u 200 ) + β ( u 212 , u 211 , u 210 ) , i.e., the v ecto r of v alues of the f ield u at the p oin ts of the line { (2 , 2 , 2) , (2 , 2 , 1 ) , ( 2 , 2 , 0) } is a linear com b ination of the v ectors of v alues of the f ield u at the p oin ts of the t w o sh aded lines { (2 , 0 , 2) , (2 , 0 , 1) , (2 , 0 , 0) } and { (2 , 1 , 2) , (2 , 1 , 1) , (2 , 1 , 0) } , and hence we can sh ade also the line { (2 , 2 , 2) , (2 , 2 , 1) , (2 , 2 , 0) } in our cub e. Th us, th e v alues of the f ie ld u are determined at all p oint s of our cub e, and all lines of the cub e are sh aded now. T he th eorem is prov ed.  Moreo v er, w e hav e pro v ed a c onsider ably str onger pr inciple of c onsistency on the cu b ic lattic e for determinants . Theorem 2 ([2]) . F or arbitr ary generic initial data, the nonline ar discr ete e quation (4) c an b e satisfie d in a c onsistent way and simultane ously on e ach set of p oints of thr e e lines P l , 1 ≤ l ≤ 3 , of the cubic la ttic e Z 3 of the form P l = { ( i, r l , s l ) , a ≤ i ≤ a + 2 } , 1 ≤ l ≤ 3 , wher e a , r l , and s l , 1 ≤ l ≤ 3 , ar e arbitr ary fixe d inte gers ( x - typ e line s ) , as wel l as on e ach set of p oints of thr e e lines Q l , 1 ≤ l ≤ 3 , of the c u bic lattic e Z 3 of the form Q l = { ( r l , j, s l ) , a ≤ j ≤ a + 2 } , 1 ≤ l ≤ 3 , wher e a , r l , and s l , 1 ≤ l ≤ 3 , ar e arbitr ary fixe d inte gers ( y -typ e lines ) , and on e ach set of p oints of thr e e line s R l , 1 ≤ l ≤ 3 , of the cub ic lattic e Z 3 of the form R l = { ( r l , s l , k ) , a ≤ k ≤ a + 2 } , 1 ≤ l ≤ 3 , wher e a , r l , and s l , 1 ≤ l ≤ 3 , ar e arbitr ary fixe d inte gers ( z -typ e lines ) . Mor e over, in this c ase the discr ete e quation (4) wil l b e satisfie d in a c onsistent way and simultane ously on e ach set of p oints of sp e cial form lying on thr e e b ent lines S l , 1 ≤ l ≤ 3 , of the same typ e in the cubic lattic e Z 3 , for example, of the form S l = { ( a, r l , s ) , ( a + 1 , r l , s ) , ( a + 1 , r l , s + 1) } , 1 ≤ l ≤ 3 , wher e a , r l , and s , 1 ≤ l ≤ 3 , ar e arbitr ary fixe d inte g ers, of the form S l = { ( a, s, r l ) , ( a + 1 , s, r l ) , ( a + 1 , s + 1 , r l ) } , 1 ≤ l ≤ 3 , wher e a , r l , and s , 1 ≤ l ≤ 3 , ar e arbitr ary fixe d inte gers, or of the form S l = { ( r l , s, a + 1) , ( r l , s, a ) , ( r l , s + 1 , a ) } , 1 ≤ l ≤ 3 , wher e a , r l , and s , 1 ≤ l ≤ 3 , ar e arbitr ary fixe d inte gers. The follo wing principle of c onsistency on the cubic lattic e for determinants also h olds. Let us consider an arb itrary line P (b en t or unb ent ) given b y th ree arbitrary neigh b oring p oints in the cubic lattice Z 3 . W e consider an arb itrary set of thr ee lines of the cubic lattice Z 3 that are obtained from the line P b y translations in the lattic e by v ecto rs p arallel to the (one- dimensional or t w o-dimensional) space orthogonal to the line P (i.e., orthog onal to the plane or On Initial Data in th e Problem of Consistency on Cubic Latti ces for 3 × 3 Determinan ts 19 to the straigh t line of P dep ending on whether the line P is b en t or un b ent). Then , for arbitrary generic initial data, the n onlinear discrete equatio n (4) can b e satisf ied in a consisten t w a y and sim ultaneously on eac h suc h set of three lines of the cubic lat tice Z 3 . Similar prop erties of consistency on cubic lattice s h old for determinan ts of arbitrary order N ≥ 2 (see [1, 2]). Ac kno wledgemen ts The w ork w as carried out under partial f inancial supp ort fr om the Russian F o undation for Basic Researc h (pro ject n o. 09-01-00 762) and from the p rogramme “Leading Scientif ic Schools” (pro ject no. NSh-5413.20 10.1). References [1] Mokho v O.I., On consistency of determinants on cub ic lattices, Usp ekhi Mat. Nauk 63 (2008), no. 6, 169–170 (English transl.: Rus sian Math. Surveys 63 (2008), no. 6, 1146–114 8), arXiv:0809.203 2 . [2] Mokho v O.I., Consistency on cubic lattices for determinants of arbitrary orders, in Geometry , T op ology and Mathematical Physics, T r. Mat. I nst. Steklova 266 (2009), 202–217 (English transl.: Pr o c. Steklov Inst. Math. 266 (2009), 195–209), arXiv:0910.20 44 . [3] Nijhof f F.W., W alker A.J., The discrete and contin uous P ainlev ´ e VI hierarch y and the Garnier systems, Glasg. Math. J. 43A (2001), 109–123, n lin.SI /000105 4 . [4] Nijhof f F.W., Lax pair for the Adler (lattice Krichev er–No viko v ) system, Phys. L ett. A 297 (2002), 49–58, nlin.SI/0110027. [5] Bobenko A.I ., Suris Y u.B., Integra ble systems on q uad-graphs, Int. Math. R es. Not. 2002 (2002), no. 11, 573–611 , nlin.SI/0110004. [6] Adler V.E., Bobenko A.I., S uris Y u.B., Cl assif ication of integrable equations on quad-graphs. The consis- tency approac h, Comm. Math. Phys. 233 (2003), 513–543, nlin.SI/0202024. [7] Bobenko A.I., Suris Y u.B., Discrete dif ferential geometry . Integrable structure, Gr aduate Studies in Mathe- matics , V ol. 98, American Mathematical So ciety , Providence, R I, 2008, math.DG/0504358. [8] Bobenko A.I., Suris Y u.B., On organizing principles of discrete dif ferential geometry . Geometry of spheres, Usp ekhi Mat. Nauk 62 (2007), no. 1, 3–50 (English transl.: Russian M ath. Surveys 62 (2007), no. 1, 1–43 ), math.DG/060829 1 . [9] V eselo v A.P ., Integr able maps, Usp ekhi Mat. Nauk 46 ( 1991), no. 5, 3–45 (En glish transl.: Rus sian Math. Surveys 46 (1991), no. 5, 1–51). [10] Adler V .E., Bob enko A .I., S uris Y u.B., Discrete nonlinear hyperb olic equations: classif ication of integra ble cases, F unct. Anal. A ppl. 43 (2009), 3–17, [11] Tsarev S.P ., W olf Th., Classif ication of three-dimensional integrable scalar discrete equations, L et t. Math. Phys 84 (2008), 31–39, [12] Hietarin ta J., A new tw o- d imensional lattice mo del that is ‘consistent around a cub e’, J. Phys. A: Math. Gen. 37 (2004), L67–L73, nlin.SI/0311034 . [13] Adler V.E., V eselo v A.P ., Cauc hy problem f or i ntegrable d iscrete equations on quad- graphs, A ct a Appl. M ath. 84 (2004), 237–262, math-ph /021105 4. [14] Adler V.E., Suris Y u.B., Q 4 : integrable master equation related to an elliptic curv e, Int. Math. R es. Not. 2004 (2004), no. 47, 2523–25 53, nlin.SI/0309030 .