On the structure of the set of reversible cellular automata

We investigate the structure of the set of reversible cellular automata. In the class of two-dimensional binary linear cellular automata with variable structure, the classes with decidable and undecidable reversibility property are separated, which contain almost all such cellular automata. With the use of this result, we prove undecidability of reversibility in the class of cellular automata with $\Gamma$-pattern and sixteen states of a cell and in the class of two-dimensional binary cellular automata with self-dual local transition functions. The complete description is given for the structure of the set of reversible cellular automata of the class of binary cellular automata with local transition functions from the Post classes.