Replying to New Zealand's total of 350, both batsmen moved on to 27 not out, with Cook five runs short of becoming England's highest test run-scorer.
The New Zealand pace bowlers kept a tight line and there were few bad balls on offer so the England openers played watchfully compared with the flamboyant stroke-play of the touring side on Thursday.
Lyth steered Tim Southee to third man for his fourth boundary to raise the 50 partnership just before the interval.
Earlier, the New Zealand tail wagged merrily after resuming on 297 for eight, adding 53 runs in half an hour.
Matt Henry was caught by wicketkeeper Jos Buttler for 27 before Mark Craig and Trent Boult added 40 for the last wicket with a few lusty blows.
Boult, on 15, skied Broad to Lyth at cover to end the innings, leaving Craig unbeaten on 41 and giving the England pace bowler figures of five for 109. – Reuters
Original source: Cook closing in on record