Leeds - BJ Watling became the first New Zealand batsman to score a Test century at Headingley as the tourists eyed a series-levelling victory over England on Sunday.
Watling's 100 not out took New Zealand to 338 for six in their second innings at stumps on the third day, with their lead also 338 after both sides made first innings totals of exactly 350.
It meant England would have to rewrite the record books were they to win this match and take the series 2-0 after their 124-run victory in the first Test at Lord's.
The most any England side have made to win in the fourth innings of a Test is the 332 for seven they posted against Australia at Melbourne back in 1928/29.
And only two sides have scored more than 300 to win in the fourth innings of a Test at Headingley.
Australia, thanks to brilliant centuries by Arthur Morris (182) and Don Bradman (173 not out) made 404 for three for a celebrated seven-wicket victory over England at Yorkshire's headquarters in 1948.
And England themselves scored 315 for four, with Mark Butcher making an unbeaten 173, to beat Australia by six wickets in 2001.
Stuart Broad proved a thorn in the side of New Zealand with both bat and ball early on the third day.
Broad ensured first innings parity and then reduced New Zealand to 23 for two in their second innings.
But in what is now typical New Zealand style, Martin Guptill (70) and Ross Taylor (48) added 99 in 87 balls for the third wicket.
That was the first of several fine stands in the innings, with Watling adding 121 with New Zealand captain Brendon McCullum (55) and 53 with Luke Ronchi (31).
South Africa-born Watling, playing as a batsman only in this match after he injured his knee keeping wicket at Lord's, went to his hundred in just under three hours off 136 balls, including 13 fours and a six, when he hooked James Anderson down to long leg for a single.
It was the 29-year-old Watling's fifth hundred in just 31 Tests and his first against England.
Guptill paid tribute to Watling by telling Sky Sports: "He is a gritty little character, he can bat anywhere in the order.
"When Watling gets going he can really get big scores."
Looking ahead, Guptill said New Zealand were already well-placed with two days left in the match.
"There is still a bit of up and down movement. It can only be a good sign for us as we have to bowl last."
Broad's 46 helped England make 350.
He then took two wickets for four runs in 12 balls, with Tom Latham and Kane Williamson both caught behind.
England bowling coach Ottis Gibson said New Zealand had hit the home side's attack of their length.
"The New Zealand batsmen have been attacking...They haven't allowed us to control our lengths," Gibson explained.
"You can't have six slips if you aren't bowling enough balls in the right area," the former West Indies paceman added.
Guptill and Ross Taylor (48) got New Zealand going again after Broad's early double strike.
But the pendulum swung back in England's favour when Taylor chipped Durham quick Mark Wood to Ben Stokes at extra-cover.
Wood was again rewarded for pitching the ball up when Guptill's intended drive flew quickly to Joe Root, who held a sharp catch at third slip.
But runs continued to come quickly as both Watling and McCullum made 79-ball fifties.
England needed to hold every chance but wicket-keeper Jos Buttler dropped a tough, rising catch when Watling gloved Ali on 50.
Next ball, Watling slog-swept Ali for six.
McCullum became only the second New Zealand player, after retired former captain Stephen Fleming, to score 6,000 Test runs when a drive off Root beat the diving Wood at mid-off and went for four.
He was eventually lbw to Wood.
England resumed on Sunday on 253 for five only to see their middle order collapse as Tim Southee took three wickets for six runs in 17 balls.
But Broad led a revival that saw England's final two wickets add 83 runs before he was last man out.