Deautonomizations of integrable equations and their reductions

We present a deautonomization procedure for partial difference and differential-difference equations (with the latter defining symmetries of the former) which uses the integrability conditions as integrability detector. This procedure is applied to Hirota’s Korteweg–de Vries and all the ABS equations and leads to non-autonomous equations and their non-autonomous generalized symmetries of order two, all of which depend on arbitrary periodic functions and are related to the same two-quad equation and its symmetries. We show how reductions of the derived differential-difference equations lead to alternating QRT maps, and periodic reductions of the difference equations result to non-autonomous maps and discrete Painlevé type equations.