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...tion [11] and its modification given in Section 5 defines a continuous symmetry of the discrete Tzitzeica equation. The role of such symmetries in the theory of quad-equations is well known, see e.g. =-=[12]-=-. 2 Linear problems Integrability of equation (3) is based on the zero curvature representation which can be conveniently written as a system of second order difference equations. Statement 1. The dis...

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...re able to find the commutators among the two sub-algebras and, thus, to describe its symmetry algebra. From the interpretation of symmetries of the ABS equations as differential-difference equations =-=[12, 18]-=- follows that they correspond to particular cases of Yamilov’s discrete Krichever-Novikov (YdKN) equation, or equation V4 with ν = 0 in Yamilov’s terminology [35]. This equation belongs to the class o...

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...(Q4) [24]. The symmetries for the discrete equations of the ABS list have been constructed [28, 27] and are given by D∆E, subcases of Yamilov’s discretization of the Krichever–Novikov equation (YdKN) =-=[17, 31]-=-: du0 d = R(u1, u0, u−1) u1 − u−1 , R(u1, u0, u−1) = A0u1u−1 +B0(u1 + u−1) + C0, where A0 = c1u 2 0 + 2c2u0 + c3, B0 = c2u 2 0 + c4u0 + c5, C0 = c3u 2 0 + 2c5u0 + c6. It is immediate to see that by d...

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...these lattice equations cannot be considered yet complete and new results which help the understanding of the interrelations between them and some differential-difference equations can still be found =-=[9]-=-. A two-dimensional partial difference equation is a functional relation among the values of a function u : Z × Z → C at different points of the lattice of indices i, j. It involves the independent va...

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...these lattice equations cannot be considered yet complete and new results which help the understanding of the interrelations between them and some differential-difference equations can still be found =-=[9]-=-. A two-dimensional partial difference equation is a functional relation among the values of a function u : Z × Z → C at different points of the lattice of indices i, j. It involves the independent va...