For India, the silver lining was Shreyas Iyer bringing up his maiden Test hundred, top-scoring with 105 while Ravichandran Ashwin played a counter-attacking knock of 38 at number eight apart from half-centuries by Ravindra Jadeja and Shubman Gill.
Resuming from 258/4, Iyer began where he left off on day one, with two boundaries off Jamieson's first over of the day. From the other end, Southee began his show of eleven straight overs, breaking the 121-run stand in his second over as Jadeja chopped on to his stumps with no feet movement, departing without adding to his overnight score.
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But Iyer was unperturbed, taking another boundary by steering through point off Jamieson.
Iyer continued to do well against Jamieson, getting boundaries through a thick edge between slip and wide gully fielder followed by an elegant cover drive.
Iyer reached his maiden Test century with two runs through point off Jamieson in the 92nd over. The 26-year-old became the 16th Indian batter (tenth at home) to score a century on Test debut.
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He also became the second Indian batter after Gundappa Viswanath (against Australia in 1969) to score a Test hundred on debut in Kanpur.
After Wriddhiman Saha nicked behind to keeper Tom Blundell off Southee in the 93rd over, Ravichandran Ashwin took consecutive boundaries off him.
Post drinks break, Southee had his third wicket of the day as Iyer chipped a drive away from the body to cover. Southee completed his 13th five-wicket haul in Test cricket, getting one to seam away from Axar Patel, drawing him into a poke and nicked behind to keeper.
Ashwin continued his counter-attack, taking a boundary each off Southee and Jamieson. He, along with Yadav, ensured that India didn't lose any more wickets leading into lunch.
Post lunch, Ashwin was bowled by Ajaz Patel, beaten by a ball turning sharply from outside leg-stump.
Patel got his second wicket of the match by trapping Ishant Sharma plumb lbw. Sharma took the review but replays showed ball hitting top of middle and leg stump, completing a comeback by New Zealand picking last six wickets for 87 runs.
Brief Scores: India 345 all out in 111.1 overs (Shreyas Iyer 105, Shubman Gill 52, Ravindra Jadeja 50, Tim Southee 5/69; Kyle Jamieson 3/85) vs New Zealand.