At what time of the year, just after the full moon, does the moon rise at almost the same time every day?

During the September full moon, the declination of the Moon increases rapidly and therefore increases its visibility over the horizon, which almost compensates for the lag of the Moon caused by its direct movement to the east.
The opposite picture is observed in March, when the declination of the full moon decreases rapidly. The moon rises every day 1 hour and 40 minutes later.

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