Are there trained neural networks, that can distinguish a book's author point from what he stands against? Let's take some specific book to narrow this example. Atlas shrugged by Ayn Rand. When i'll be saying something like "Ayn Rand's ideas", i'll mean only those, which are clearly stated in the book.  

Humans who read this book, can clearly distinguish sentences, that describe Ayn Rand's point of view from sentences, that oppose it.

Are there trained neural networks, that could distinguish sentences with Ayn Rand's ideas vs sentences with opposing ideas?

If not, are there any reasons, why it's impossible to train such network, if each sentence in such book would be marked as pro-, contra- or neutral to Ayn Rand's ideas? Successfully trained network should be able to distinguish author's point of view in books, where humans can clearly do it.
Absolutely. You've described topic modeling and sentiment analysis. You might want to train separate models for each part of the problem and stack them, but this is a totally solvable problem.