In the following sentence: " He likes to sit there and read a book". I've been noticing that most of the time, we don't use 'to' after the word 'and', but even though there isn't 'to' before the verb, it is still in the infinitive form. Is there any rule which explains that?
CGEL talks about the nestable "head"--"constituent" structure that English uses for many things.
If you connect multiple structures with and, and all such structures have the same "head", the head often only needs to be specified once, especially if the structures are single words or short. The head can optionally be repeated for clarity.
Things like articles, the first verb do, be, or have in auxillary verb phrases, and the to in infinitive expressions are all types of head elements.
I did talk to Mary and tell her to leave (don't need to repeat "did").
I have stopped and started the engine again (no need to repeat "have").
He wasn't able to talk and chew gum at the same time (no need to repeat "to).