Ben Stokes’ side are aiming to finish the year on a high against the Black Caps, who come into the series on the back of an impressive 3-0 win in India.
The last series between England and New Zealand two years ago was drawn, while England’s last win in the country was in 2008.
The three-match series starts in Christchurch on November 28.
It is followed by Tests in Wellington and Hamilton.
Play starts at 10pm UK time each day for all three Test matches.
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Bethell made his first England appearances this summer, featuring in five one-day internationals as well as two T20s against Australia.
He has yet to register a century and holds a batting average of 25.44, but made an impressive start to life in an England shirt in the summer’s T20I and ODI action – also providing variation with left-arm spin.
Meanwhile, Cox is set to deputise for the absent Smith. The debutant has appeared in two T20s while also being the back-up batter and keeper in the Test squad since England’s series against Sri Lanka in the summer.
McCullum has opted to retain the experience of Ollie Pope as vice-captain despite a disappointing average of 11 in the defeat to Pakistan earlier this month.
New Zealand bowler Tim Southee will retire following the series. Southee, 35, has claimed 385 Test wickets in 104 appearances.
New Zealand’s leading Test run-scorer Kane Williamson is back in the squad after missing the series win in India with injury.
Wellington all-rounder Nathan Smith has been handed his first Test call-up and seamer Jacob Duffy is also part of the squad. Spinner Mitchell Santner is available for the second and third Tests.
New Zealand: Tom Latham, Tom Blundell, Devon Conway, Jacob Duffy, Matt Henry, Daryl Mitchell, Will O’Rourke, Glenn Phillips, Rachin Ravindra, Mitchell Santner (Tests 2 and 3), Nathan Smith, Tim Southee, Kane Williamson, Will Young
England: Ben Stokes, Rehan Ahmed, Gus Atkinson, Shoaib Bashir, Jacob Bethell, Harry Brook, Bryson Carse, Jordan Cox, Zak Crawley, Ben Duckett, Jack Leach, Ollie Pope, Matthew Potts, Joe Root, Olly Stone, Chris Woakes.
Joe Root has been recalled to the England ODI side for the first time since the Jos Buttler-captained team put up an underwhelming ODI World Cup defence in Nove
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