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Southampton were much the better side in the first half as in-form striker Rickie Lambert hit the woodwork with a bending free kick and narrowly failed to score with two trademark headers.
Everton improved in the second half but also failed to convert chances as the home side clung on for a valuable draw.
Everton moved to 38 points, within three of fourth-placed Tottenham Hotspur and four ahead of Arsenal and Liverpool while Southampton edged further away from the relegation zone to 23 points, four above the drop.
Former Argentina international Pochettino was controversially appointed manager despite Southampton's improved recent form, including a 2-2 comeback draw from two goals down at European champions Chelsea on Wednesday, Adkins's final game in charge.