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As @DanielVassallo explained it, retrieving the last day of next month is easy:
SELECT LAST_DAY(DATE_ADD(CURRENT_DATE(), INTERVAL 1 MONTH));
To retrieve the first day, you could first define a custom FIRST_DAY function (unfortunately MySQL does not provide any):
DELIMITER ;;
CREATE FUNCTION FIRST_DAY(day DATE)
RETURNS DATE DETERMINISTIC
BEGIN
RETURN ADDDATE(LAST_DAY(SUBDATE(day, INTERVAL 1 MONTH)), 1);
END;;
DELIMITER ;
And then you could do:
SELECT FIRST_DAY(DATE_ADD(CURRENT_DATE(), INTERVAL 1 MONTH));