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...of the studies on these equations presented in [15], [27], [28], [30]. The effectiveness of using Lie point symmetries of integrable lattice equations in obtaining YangBaxter maps was demonstrated in =-=[23]-=-, [24]. In particular, it was shown there that the Yang-Baxter variables can be chosen as invariants of the multi-parameter symmetry groups of the integrable lattice equations. Here, it was shown that...

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...the Kulish system, because they do not respect the reduction to osp(1|2) superalgebra. The results obtained here can be developed in several directions: 1) to study the corresponding Yang-Baxter maps =-=[20, 39]-=-; 2) To derive the recursion operators and study the associates multi-Hamiltonian structures; 3) This can be generalised to other integrable hierarchies. Acknowledgements The authors have the pleasure...

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...Baxter maps on CP1 × CP1 is given in [1] and [2] respectively (see also [14]). Higher dimensional Yang-Baxter maps are obtained from multifield integrable lattice equations through symmetry reduction =-=[15, 16]-=-. Loop groups equipped with the Sklyanin bracket provide a natural framework in order to derive Yang-Baxter maps with polynomial Lax matrices. In [17] one of the most fundamental examples of a paramet...

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...ults of YB maps for X = CP 1 , in connection to analogous results for two–dimensional integrable discrete equations on the square lattice [5, 6]. The latter correspondence was further investigated in =-=[26, 27]-=- by exploiting the local groups of symmetry transformations of the discrete equations. It was demonstrated that an integrable quadrilateral equation with a sufficient r-parameter symmetry group gives ...

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...riables f are associated with the edges of the lattice rather than the vertices, and in this situation one should use another definition which generalizes the notion of Yang–Baxter mappings, see e.g. =-=[22, 23, 24, 25]-=-. Certainly, both versions of 3D-consistency, for the vertices and the edges, are closely related. It turns out that equations (7) admit introducing of the potential (due to some conservation law) u =...

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..., y2, x1, x2 + a− b 1 + x1y2 y2 ) . (5.17) Map (5.17) coincides with the Adler-Yamilov map. ✷ Map (5.17) originally appeared in the work of Adler and Yamilov [10]. Moreover, it appears as a YB map in =-=[51, 76]-=-. Now, one can use the condition X = x1x2+a to eliminate X from the Lax matrix (5.12), i.e. M(x; a, λ) = λ 1 0 0 0 + a + x1x2 x1 x2 1 , x = (x1, x2). (5.18) The form of Lax matrix (5.18) ...