Record on class in date and time

Whether it is possible to make so that the user after registration could register in date and time of occupation and after that occupation in this date and time it wasn't available to record any more?

Sorry, I did not understand, what's your problem. Can you explain a bit more, please!

The user after registration on the website can register in occupation to the teacher for a certain date and time? The component allows to make such functionality?

sorry, I must understand your circumstances to understand, what you want.

I want to learn a possibility of realization of the necessary functionality.

The user can register in actions to a certain teacher?
For concrete date and time?

As much as I understand you, this is not possible.
You can create events e.g. lessons of a teacher with given time
It's is possible too to create events (lessons) with no time.
Then a user can signup for an event e.g. lesson of a teacher. But each event e.g. lesson is standalone.
For lessons with no time, the user could write in the optional comment what time he wishes. But then the event creator must do handwork!