1. followed by the non-perfect common aspect infinitive expresses
1.1. Physical and mental ability or capacity
1.1.1. The notion of ability is also expressed by “to be able to”
1.2. Possibility
1.2.1. possibility due to circumstances
1.2.2. possibility due to the existing rules of laws
1.2.3. possibility of the idea (the so-called “theoretical” possibility)
1.3. Permission
1.3.1. Can is now more common than may (or might)
1.4. Prohibition
1.4.1. only with the negative form of the modal verb
1.5. Request
1.5.1. Could suggests a greater degree of politeness
2. followed by any form of the infinitive may express
2.1. Strong doubt, improbability, incredulity
2.1.1. only with the negative form of the modal verb + perfect infinitive, continuous infinitive, or be.
2.1.2. could showing a greater degree of doubt or incredulity
2.2. Surprise
2.2.1. when can/could is used in questions. It corresponds to the Russian неужели
2.2.2. То refer the action to the past a perfect infinitive is used.
2.2.3. Is used in negative form only in questions
2.2.3.1. by complex sentences: Can if be that you haven’t seen him? Неужели вы не видели его?
2.2.3.2. by different lexical means: Can you have failed to see him? Неужели вы не видели его?
2.3. Reproach
2.3.1. implying that a person should have done something, or behaved in a certain way, but didn’t do it.
2.3.2. only with the form could.
2.4. Purpose
2.4.1. only with the form could in clauses of purpose.