Captain Taylor reached the close on 119 not out and Williamson was on 95 as the pair put on 209 and counting.
It helped the tourists recover in style from a shaky start, the hosts — fresh from their thumping 10-wicket win in the first Test — having reduced their opponents to 14 for two in the fourth over.
The momentum then shifted dramatically, though, and Taylor with 10 boundaries and Williamson with eight put their side in a position of real authority at 223 for two.
Openers Martin Guptill and Brendon McCullum both fell for four as New Zealand got off to a terrible start.
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