We show that integrable involutive maps, due to the fact they admit three integrals in separated form, can give rise to equations, which are consistent around the cube and which are not in the multiaffine form assumed in papers [1,2]. Lattice models, which are discussed here, are related to the lattice potential KdV equation by nonlocal transformations (discrete quadratures).

A relationship between the tetrahedron equation for maps and the consistency property of integrable discrete equations on 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.

Abstract. We present a method to obtain families of lattice equations. Specifically we focus on two of such families, which include 3-parameters and their members are connected through Bäcklund transformations. At least one of the members of each family is integrable, hence the whole family inherits some integrability properties.

We present the explicit form of a family of Liouville integrable maps in 3 variables, the so-called triad family of maps and we propose a multi-field generalisation of the latter. We show that by imposing separability of variables to the invariants of this family of maps, the H I , H II and H A III Yang-Baxter maps in general position of singularities emerge. Two different methods to obtain entwining Yang-Baxter maps are also presented. The outcomes of the first method are entwining maps associated with the H I , H II and H A III Yang-Baxter maps, whereas by the second method we obtain non-periodic entwining maps associated with the whole F and H-list of quadrirational Yang-Baxter maps. Finally, we show how the transfer maps associated with the H-list of Yang-Baxter maps can be considered as the (k − 1)-iteration of some maps of simpler form. We refer to these maps as extended transfer maps and in turn they lead to k-point alternating recurrences which can be considered as alternating versions of some hierarchies of discrete Painlevé equations.

Abstract. Given a constant skew-symmetric matrix A, it is a difficult open problem whether the associated Lotka-Volterra system is integrable or not. We solve this problem in the special case when A is a Toepliz matrix where all off-diagonal entries are plus or minus one. In this case, the associated Lotka-Volterra system turns out to be a reduction of Liouville integrable systems, whose integrability was shown by Bogoyavlenskij and Itoh. We prove that the reduced systems are also Liouville integrable and that they are also non-commutative integrable by constructing a set of independent first integrals, having the required involutive properties (with respect to the Poisson bracket). These first integrals fall into two categories. One set consists of polynomial functions which can be obtained by a matricial reformulation of Itoh's combinatorial description. The other set consists of rational functions which are obtained through a Poisson map from the first integrals of some recently discovered superintegrable Lotka-Volterra systems. The fact that these polynomial and rational first integrals, combined, have the required properties for Liouville and non-commutative integrability is quite remarkable; the quite technical proof of functional independence of the first integrals is given in detail.