On a nonlinear integrable difference equation on the square 3D-inconsistent

We present a nonlinear partial difference equation defined on a square which is obtained by combining the Miura transformations between the Volterra and the modified Volterra differential-difference equations. This equation is not symmetric with respect to the exchange of the two discrete variables and does not satisfy the 3D-consistency condition necessary to belong to the Adler-Bobenko-Suris classification. Its integrability is proved by constructing its Lax pair. The uncovery of new nonlinear integrable completely discrete equations is always a very challenging problem as, by proper continuous limits, many other results on differential-difference and partial differential equations can be obtained. In the case of differential equations by now a lot is known starting from the pioneering works by Gardner, Green, Kruskal and Miura. A summary of these results is already of public domain and presented for example in the Encyclopedia of Mathematical Physics [5] or in the Encyclopedia of Nonlinear Science [6]. Among those results let us mention the classification scheme of nonlinear integrable partial differential equations introduced by Shabat using the formal symmetry approach, see [11] for a review. The classification of differential-difference equations has also been carried out using the formal symmetry approach by Yamilov [17] and it is a well defined procedure which can be easily computerized for many families of equations [10,18].