You can get the same effect in mostly the same situations by using the pro-verb "do so" instead of ellipsis/auxiliary stranding:
Both forms of reduction appear to require the gap or the pro-form to replace not just the verb, but also all its complements. However, adjuncts such as "at the store" and "on Sunday" can be still be attached.
For more information on these constructions, see Cambridge Grammar of the English Language, ch. 17, section 7, especially §7.1 (auxiliary stranding) and §7.5 (do so).