We classify rational solutions of a specific type of the set theoretical version of the pentagon equation. That is, we find all quadrirational maps R : (x , y) -> (u(x,y), v(x, y)), where u , v are two rational functions on two arguments, that serve as solutions of the pentagon equation. Furthermore, provided a pentagon map that admits a partial inverse, we obtain genuine entwining pentagon set theoretical solutions. Finally, we show how to obtain Yang-Baxter maps from entwining pentagon maps.

We present two lists of multi-component systems of integrable difference equations defined on the edges of a $\mathbb{Z}^2$ graph. The integrability of these systems is manifested by their Lax formulation which is a consequence of the multi-dimensional compatibility of these systems. Imposing constraints consistent with the systems of difference equations, we recover known integrable quad-equations including the discrete version of the Krichever-Novikov equation. The systems of difference equations allow us for a straightforward reformulation as Yang-Baxter maps. Certain two-component systems of equation defined on the vertices of a $\mathbb{Z}^2$ lattice, their non-potential form and integrable equations defined on 5-point stencils, are also obtained.

A relationship between the tetrahedron equation for maps and the consistency property of integrable discrete equations on $\mathbb{Z}^3$ is investigated. Our approach is a generalization of a method developed in the context of Yang-Baxter maps, based on the invariants of symmetry groups of the lattice equations. The method is demonstrated by a case-by-case analysis of the octahedron type lattice equations classified recently, leading to some new examples of tetrahedron maps and integrable coupled lattice equations.

We use the classification of the quadrirational maps given by Adler, Bobenko and Suris to describe when such maps satisfy the Yang-Baxter relation. We show that the corresponding maps can be characterized by certain singularity invariance condition. This leads to some new families of Yang-Baxter maps corresponding to the geometric symmetries of pencils of quadrics.

We present Yang-Baxter maps associated to elliptic curves. They are related to discrete versions of the Krichever-Novikov and the Landau-Lifshits equations. A lifting of scalar integrable quad-graph equations to two-field equations is also shown.

We consider lattice equations on Z 2 which are autonomous, affine linear and possess the symmetries of the square. Some basic properties of equations of this type are derived, as well as a sufficient linearization condition and a conservation law. A systematic analysis of the Lie point and the generalized three-and five-point symmetries is presented. It leads to the generic form of the symmetry generators of all the equations in this class, which satisfy a certain non-degeneracy condition. Finally, symmetry reductions of certain lattice equations to discrete analogues of the Painlevé equations are considered.

A variety of set-theoretic solutions to the quantum Yang-Baxter relation are obtained from integrable multi-field equations on quad-graphs. A systematic framework for investigating this connection relies on the symmetry groups of the equations. The method is applied to lattice equations introduced by Adler and Yamilov and which are related to the nonlinear superposition formulae for the Bäcklund transformations of the nonlinear Schrödinger system and specific ferromagnetic models.

We consider lattice equations on Z which are autonomous, affine linear and possess the symmetries of the square. Some basic properties of equations of this type are derived, as well as a sufficient linearization condition and a conservation law. A systematic analysis of the Lie point and the generalized threeand five-point symmetries is presented. It leads to the generic form of the symmetry generators of all the equations in this class, which satisfy a certain non-degeneracy condition. Finally, symmetry reductions of certain lattice equations to discrete analogs of the Painlevé equations are considered. PACS number: 02.20.Sv Mathematics Subject Classification: 39A05, 70G65

We consider lattice equations on Z(2) which are autonomous, affine linear and possess the symmetries of the square. Some basic properties of equations of this type are derived, as well as a sufficient linearization condition and a conservation law. A systematic analysis of the Lie point and the generalized three- and five-point symmetries is presented. It leads to the generic form of the symmetry generators of all the equations in this class, which satisfy a certain non-degeneracy condition. Finally, symmetry reductions of certain lattice equations to discrete analogs of the Painleve equations are considered.

A variety of Yang–Baxter maps are obtained from integrable multi-field equations on quad-graphs. A systematic framework for investigating this connection relies on the symmetry groups of the equations. The method is applied to lattice equations introduced by Adler and Yamilov, which are related to the nonlinear superposition formulae for the Bäcklund transformations of the nonlinear Schrödinger system and specific ferromagnetic models.

The study of Sophus Lie in the late nineteenth century on the unification and extension of various solution methods for ordinary differential equations led him to introduce the notion of continuous groups of symmetry transformations. During the same period Backlund investigated possible extensions of Lie contact transformations, introducing an important class of surface transformations in ordinary space. A remarkable feature of Backlund transformations is that due to a commutativity property repeated applications can be performed in a purely algebraic manner. This is known in classical geometry as the Bianchi permutability theorem and represents a nonlinear analogue of the superposition principle for linear homogeneous differential equations. The archetype is given by the equation

A symmetry reduction of the lattice modified Boussinesq system is studied. The full group of Lie point symmetries of the relevant system is retrieved and certain group invariant solutions are considered by using an accessional generalized symmetry. It is demonstrated that the symmetry reduction leads to a coupled set of second-order nonlinear non-autonomous ordinary difference equations involving six free parameters, generalizing to higher order some of the known discrete analogues of the Painlevé VI equation. The corresponding isomonodromic deformation problem is constructed through the symmetry reduction as well.

A connection between the Yang-Baxter relation for maps and the multidimensional consistency property of integrable equations on quad-graphs is investigated. The approach is based on the symmetry analysis of the corresponding equations. It is shown that the Yang-Baxter variables can be chosen as invariants of the multiparameter symmetry groups of the equations. We use the classification results by Adler, Bobenko, and Suris to demonstrate this method. Some new examples of Yang-Baxter maps are derived in this way from multifield integrable equations.

It is shown that the symmetries of a partial differential equation which incorporates the hyperbolic Ernst equation in general relativity, generate the complete hierarchy of the Korteweg-de Vries soliton equations.

A new family of nonlinear partial differential equations is presented. They represent a generalization of the hyperbolic Ernst equations for an Einstein-Mawxell-Weyl field in general relativity. A Backlund transformation for the system of equations under consideration is given, and their direct relation to the complete Boussinesq hierarchy of soliton equations is illustrated.

We consider the discrete Boussinesq integrable system and the compatible set of differential difference, and partial differential equations. The latter not only encode the complete hierarchy of the Boussinesq equation, but also incorporate the hyperbolic Ernst equations for an Einstein-Maxwell-Weyl field in general relativity. We demonstrate a specific symmetry reduction of the partial differential equations, to a six-parameter, second order coupled system of ordinary differential equations, which is conjectured to be of Garnier type.

We show how to construct four-dimensional anti-self-dual conformal structures from solutions of a parameter family of partial differential equations which generalize the Ernst equation of General Relativity. Anti-self-dual vacuum metrics may be obtained as well by restricting appropriately the values of the parameters.

We present a reduction of the anti-self-dual Yang–Mills (ASDYM) equations to a system of partial differential equations (PDEs) introduced recently by Nijhoff et al. (Phys. Lett. A 267 (2000) 147). An auto-Bäcklund transformation of the reduced system is also presented. The system under consideration is related to a fourth-order nonlinear PDE of the Schwarzian type. The symmetry group of the latter equation is calculated and similarity reductions to the Schwarzian equation and the full Painlevé III, V and VI are presented.

A new system of integrable nonlinear equations of hyperbolic type, obtained by a two-dimensional reduction of the anti-self-dual Yang–Mills equations, is presented. It represents a generalization of the Ernst–Weyl equation of General Relativity related to colliding neutrino and gravitational waves, as well as of the fourth order equation of Schwarzian type related to the KdV hierarchies, which was introduced by Nijhoff, Hone, and Joshi recently. An auto-Bäcklund transformation of the new system is constructed, leading to a superposition principle remarkably similar to the one connecting four solutions of the KdV equation. At the level of the Ernst–Weyl equation, this Bäcklund transformation and the associated superposition principle yield directly a generalization of the single and double Harrison transformations of the Ernst equation, respectively. The very method of construction also allows for revealing, in an essentially algorithmic fashion, other integrability features of the main subsystems, such as their reduction to the Painlevé transcendents.