Physical or mental ability. In this meaning it can be used only with the Indefinite Infinitive.

Cousin Val can’t walk much, you know, but he can ride perfectly. He’s going to show me the gallops. (Galsworthy)
I don’t know what Captain Loomis was doing there, but you can guess — it was about Thorpe. (Heym)

#1. Due to circumstances:

I could not go to the theatre yesterday because I was busy.
“Can I have a cup of tea, Miss?” she asked. But the waitress went on doing her hair. “Oh,” she sang, “we’re not open yet.” (Mansfield)
(Here the speaker does not ask the waitress’s permission to have a cup of tea but she merely wants to know whether the place is open and whether she can have a cup of tea.)
You can have the book when I have finished it.
“Can I see him?” Yates asked, not very hopefully. “I’m sorry, no,” the voice was hesitant. “We don’t want him disturbed.” (Heym)
Can I have my photo taken?

In the last three sentences can comes close to may expressing permission, but it does not so much indicate possibility depending upon the will of the person addressed (which is the essence of permission) as possibility depending on circumstances.

“Well, I’ll just wait a moment, if I may.” “You can’t wait in the waiting- room, Miss. I haven’t done it yet.” (Mansfield)

#2. due to the existing laws:

The more she studied, the less sure she became, till idly turning the pages, she came to Scotland. People could be married there without any of this nonsense. (Galsworthy)
Every Soviet citizen can get medical treatment free of charge. You cannot play football in the street (i. e. you have no right).

Can expressing possibility is used with the Indefinite Infinitive only.

Incredulity, doubt, astonishment (they are closely connected).

In these meanings can is used with all the forms of the Infinitive in interrogative and negative sentences, though astonishment is expressed only in interrogative sentences.

No wonder her father had hidden that photograph… But could he hate Jon’s mother and yet keep her photograph? (Galsworthy)
“That is not true!” exclaimed Linton, rising in agitation. “It cannot be; it is incredible, it cannot be.” (E. Bronte)
Can she be waiting for us? She cannot be waiting for us.
Can (could) she have said that?
There could not have been such relentless unforgiveness. (Galsworthy)
“Oh!” cried Fleur. “You could not have done it.” (Galsworthy)
Can she have been waiting for us all the time? She cannot have been waiting for us all the time.
Confusion came on Jon’s spirit. How could she say such things, just as they were going to part? (Galsworthy)
What could she have seen in that fellow Bosinney to send her mad? (Galsworthy)
B r e t t. They saw us. I hope they didn’t recognize us. G e n e v r a. The car was going too fast. They couldn’t have seen us. (Gow and D’Usseau)

In the last two examples we can easily replace could by can without any change of meaning.

Could is often used in polite requests.

Could you give us a consultation tomorrow?

Could with the Perfect Infinitive has almost the same meaning as can with the Perfect Infinitive, only the negation is not so categoric as with can.

If the action refers to the past, the Perfect Infinitive is mostly used. Other forms of the Infinitive are hardly ever used.