Oughtn't (to) do

[SARDORBEK]

We know that after the modal ought we use the infinitive with to.
And this makes ought different from other modal auxiliary verbs:
You ought to see a dentist.

However, in one of the well-known American English grammar books
the following definition is given:
Ought to is not commonly used in the negative. If it is used in the negative,
the to is sometimes dropped:

You oughtn't (to) leave your keys in the car.

Dear ALAN! I have not met this kind of rule anywhere, so I have four questions regarding this
subject:

1) Is the particle to optional in all types of situations?
2) If it is true, does this rule apply only to contracted form of ought not ?
3) Is there any difference between Am.E and Br.E?
4) Also, any difference between formal and informal language?

Could YOU please answer all four questions?

Respectfully.

[Alan]

I disagree with your grammar book!

In standard usage, the 'to' is never omitted, whether the phrase is positive or negative.