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We introduce a so-called `coprimeness-preserving non-integrable' extension(another terminology is `quasi-integrable' extension) to the two-dimensionalToda lattice equation. We believe that this equation is the first example ofsuch discrete equation defined over a three-dimensional lattice. We prove thatall the iterates of the equation are irreducible Laurent polynomials of theinitial data and that every pair of two iterates is co-prime, which indicateconfined singularities of the equation. By reducing the equation to two- orone-dimensional lattices, we obtain coprimeness-preserving non-integrableextensions to the one-dimensional Toda lattice equation and the Somos-4recurrence.

Coprimeness-preserving non-integrable extension to the two-dimensional discrete Toda lattice equation