You use "next Monday or month" rather than "the next Monday or month" when you mean Monday or month after the one in the present week or month.
So, let's take a look at your example, "I am going to be busy next month." Assuming that when you say this, the month is March. Then, next month means April here regardless when you say this(any day in the first, second, third or fourth week in March).
However, you use "the next" to refer to a length of time from now into the future. With "the next week", it means the seven days from today until the same day in the week after this; "the next month" means four weeks starting from today, and "the next year" means the twelve months from now till the same time in the year after this. So, the day when you say "It is going to rain for the next month" is on Friday in January, then the next month will be the period four weeks starting from the present Friday in January to the next Friday in February.
As shown from the above, you can see that there is a delicate difference between with and without "the" before "next".
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