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They proved that if (1.1) has admissible meromorphic solutions of finite order, then either w satisfies a difference Riccati equation, or (1.1) can be transformed to a list of difference equations, which contains many integrable equations, especially the difference Painlevé I, II equations.
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Here is a proof that the rule "If ∀x (A ∨ B x)) is a theorem, so is A ∨ ∀x B x)" (where x is not free in A) is not admissible for HA, if HA is consistent.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com