Geodesic
Integrable M\"obius-invariant evolutionary lattices of second order
Funkcionalʹnyj analiz i ego priloženiâ, Tome 50 (2016) no. 4, pp. 13-25.

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We solve the classification problem for integrable lattices of the form $u_{,t}=f(u_{-2},\dots,u_2)$ under the additional assumption of invariance with respect to the group of linear-fractional transformations. The obtained list contains five equations, including three new ones. Difference Miura-type substitutions are found, which relate these equations to known polynomial lattices. We also present some classification results for generic lattices.
Keywords: integrability, symmetry, conservation law, Möbius invariantm cross-ratio. 
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V. E. Adler. Integrable M\"obius-invariant evolutionary lattices of second order. Funkcionalʹnyj analiz i ego priloženiâ, Tome 50 (2016) no. 4, pp. 13-25. http://geodesic.mathdoc.fr/item/FAA_2016_50_4_a2/

[1] K. Narita, “Soliton solution to extended Volterra equation”, J. Phys. Soc. Japan, 51:5 (1982), 1682–1685 | DOI | MR

[2] Y. Itoh, “Integrals of a Lotka–Volterra system of odd number of variables”, Progr. Theoret. Phys., 78:3 (1987), 507–510 | DOI | MR

[3] O. I. Bogoyavlensky, “Integrable discretizations of the KdV equation”, Phys. Lett. A, 134:1 (1988), 34–38 | DOI | MR

[4] O. I. Bogoyavlenskii, “Algebraicheskie konstruktsii integriruemykh dinamicheskikh sistem — rasshirenie sistemy Volterra”, UMN, 46:3 (1991), 3–48 | MR | Zbl

[5] Yu. B. Suris, The Problem of Integrable Discretization: Hamiltonian Approach, Birkhäuser, Basel, 2003 | MR | Zbl

[6] A. K. Svinin, “On some integrable lattice related by the Miura-type transformation to the Itoh–Narita–Bogoyavlenskii lattice”, J. Phys. A, 44:46 (2011), 465210 | DOI | MR | Zbl

[7] G. Berkeley, S. Igonin, “Miura-type transformations for lattice equations and Lie group actions associated with Darboux–Lax representations”, J. Phys. A, 49:27 (2016), 275201 ; arXiv: 1512.09123 | DOI | MR | Zbl

[8] X. B. Hu, P. A. Clarkson, R. Bullough, “New integrable differential-difference systems”, J. Phys. A, 30:20 (1997), L669–676 | DOI | MR

[9] V. E. Adler, V. V. Postnikov, “Differential-difference equations associated with the fractional Lax operators”, J. Phys. A, 44:41 (2011), 415203 | DOI | MR | Zbl

[10] R. N. Garifullin, R. I. Yamilov, “Generalized symmetry classification of discrete equations of a class depending on twelve parameters”, J. Phys. A, 45:34 (2012), 345205 | DOI | MR | Zbl

[11] R. N. Garifullin, A. V. Mikhailov, R. I. Yamilov, “Diskretnoe uravnenie na kvadratnoi reshetke s nestandartnoi strukturoi vysshikh simmetrii”, TMF, 180:1 (2014), 17–34 | DOI | MR | Zbl

[12] V. G. Papageorgiou, F. W. Nijhoff, “On some integrable discrete-time systems associated with the Bogoyavlensky lattices”, Phys. A, 228:1–4 (1996), 172–188 | DOI | MR

[13] F. W. Nijhoff, “On some “Schwarzian” equations and their discrete analogues”, Algebraic Aspects of Integrable Systems, Progress in Nonlinear Differential Equations and Their Applications, 26, Birkhäuser, Boston, 1997, 237–260 | MR | Zbl

[14] D. Levi, M. Petrera, C. Scimiterna, “The lattice Schwarzian KdV equation and its symmetries”, J. Phys. A, 40:42 (2007), 12753–12761 | DOI | MR | Zbl

[15] V. E. Adler, V. V. Postnikov, “On discrete 2D integrable equations of higher order”, J. Phys. A, 47:4 (2014), 045206 | DOI | MR | Zbl

[16] R. I. Yamilov, “O klassifikatsii diskretnykh evolyutsionnykh uravnenii”, UMN, 38:6 (1983), 155–156

[17] R. I. Yamilov, Diskretnye uravneniya vida $du_n/dt=F(u_{n-1},u_n,u_{n+1})$ ($n\in\mathbb{Z}$) s beskonechnym naborom lokalnykh zakonov sokhraneniya, Diss....kand. fiz.-mat. nauk, BF AN SSSR, Ufa, 1984

[18] R. I. Yamilov, “Symmetries as integrability criteria for differential difference equations”, J. Phys. A, 39:45 (2006), R541–R623 | DOI | MR | Zbl

[19] A. V. Mikhailov, A. B. Shabat, R. I. Yamilov, “Simmetriinyi podkhod k klassifikatsii nelineinykh uravnenii. Polnye spiski integriruemykh sistem”, UMN, 42:4 (1987), 3–53 | MR

[20] D. Levi, R. I. Yamilov, “Conditions for the existence of higher symmetries of evolutionary equations on the lattice”, J. Math. Phys., 38:12 (1997), 6648–6674 | DOI | MR | Zbl

[21] V. E. Adler, “Neobkhodimye usloviya integriruemosti dlya evolyutsionnykh uravnenii na reshetke”, TMF, 181:2 (2014), 276–295 | DOI | Zbl

[22] V. E. Adler, “Integrability test for evolutionary lattice equations of higher order”, J. Symbolic Comput., 74 (2016), 125–139 | DOI | MR | Zbl

[23] V. E. Adler, V. V. Postnikov, “On vector analogs of the modified Volterra lattice”, J. Phys. A, 41:45 (2008), 455203 | DOI | MR | Zbl