Have or Have got + to-infinitive
Have and have got + V-to-inf meanings
have to I have to finish these letters. (obligation)
There has to be a solution.
have got to I’ve got to go now. Oh, do you have to? (obligation)
There’s got to be a solution. (necessity)
Like must, these combinations have meanings of both obligation and necessity. With these modal meanings, have got to is more common in spoken BrE , while have to is the preferred form in AmE (Do you have to go?) Both gonna (‘going to’) and gotta (‘got to’) are informal uses.
The Subject-Finite operator inversion characteristic of be, have and other auxiliaries in interrogative and negative clauses, together with the requirement of a do operator by lexical verbs. As a reminder here, we exemplify have to and have got to in interrogative clauses, showing that while have to can function either as a primary auxiliary or as a lexical verb, have got to functions only as an auxiliary:

Furthermore, have got + to-infinitive has no non-finite forms and does not combine with modals. None of the following starred structures are possible, therefore, all being used with have to:
Incorrect Correct
*To have got to live there must be deadful To have to live there . . . dreadful
*I don’t like having got to get up early. I don’t like having to get up early.
*We have had got to repaint the kitchen. We have had to repaint the kitchen.
*You will have got to watch out for bugs there. You will have to watch out for bugs.