Darboux-integrable discrete systems

We extend Laplace's cascade method to systems of discrete “hyperbolic” equations of the form $u_{i+1,j+1}=f(u_{i+1,j},u_{i,j+1},u_{i,j})$, where $u_{ij}$ is a member of a sequence of unknown vectors, $i,j\in\mathbb Z$. We introduce the notion of a generalized Laplace invariant and the associated property of the system being “Liouville.” We prove several statements on the well-definedness of the generalized invariant and on its use in the search for solutions and integrals of the system. We give examples of discrete Liouville-type systems.