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We study the generalized symmetry structure of three known discretenonautonomous equations. One of them is the semidiscrete dressing chain ofShabat. Two others are completely discrete equations defined on the squarelattice. The first one is a discrete analogue of the dressing chain introducedby Levi and Yamilov. The second one is a nonautonomous generalization of thepotential discrete KdV equation or, in other words, the H1 equation of thewell-known Adler-Bobenko-Suris list. We demonstrate that these equations havegeneralized symmetries in both directions if and only if their coefficients,depending on the discrete variables, are periodic. The order of the simplestgeneralized symmetry in at least one direction depends on the period and may bearbitrarily high. We substantiate this picture by some theorems in the case ofsmall periods. In case of an arbitrarily large period, we show that it ispossible to construct two hierarchies of generalized symmetries andconservation laws. The same picture should take place in case of anynonautonomous equation of the Adler-Bobenko-Suris list.

Peculiar symmetry structure of some known discrete nonautonomous equations