Continuous symmetries of certain nonlinear partial difference equations and their reductions

•Lie symmetry approach is extended to nonlinear partial difference equations or lattice equations.•Five distinct nonlinear partial difference equations of QRT type admitting continuous point symmetries are reported.•Integrability nature of the partial difference equations is analyzed through degree growth of iterates.•New Third and Fourth order ordinary difference equations of QRT type are given.•How to derive higher order ordinary difference equations and their integrals is explicitly explained.

In this article, Quispel, Roberts and Thompson type of nonlinear partial difference equation with two independent variables is considered and identified five distinct nonlinear partial difference equations admitting continuous point symmetries quadratic in the dependent variable. Using the degree growth of iterates the integrability nature of the obtained nonlinear partial difference equations is discussed. It is also shown how to derive higher order ordinary difference equations from the periodic reduction of the identified nonlinear partial difference equations. The integrability nature of the obtained ordinary difference equations is investigated wherever possible.