Integrability conditions for parameterized linear difference equations

We study integrability conditions for systems of parameterized linear difference equations and related properties of linear differential algebraic groups. We show that isomonodromicity of such a system is equivalent to isomonodromicity with respect to each parameter separately under a linearly differentially closed assumption on the field of differential parameters. Due to our result, it is no longer necessary to solve non-linear differential equations to verify isomonodromicity, which will improve efficiency of computation with these systems. Moreover, it is not possible to further strengthen this result by removing the requirement on the parameters, as we show by giving a counterexample. We also discuss the relation between isomonodromicity and the properties of the associated parameterized difference Galois group.