Mail continually asks me to re enter my password

I have an i Mac running 10.7.5. I keep 2 gmail accounts and 2 AOL accounts on the Mail client. Both AOL accounts and one of the gmail accounts are IMAP, and the other gmail is POP.

Several times a day, the Mail program asks me to enter my password for some or all of the accounts. Sometimes when sending an email it will ask me as well.

I judiciously enter the password, and also check the box that says 'save to keychain'.

In addition, I have gone to Mail / Preferences / General and have noticed that the Password Box is empty. I enter the password in the box then go to the next mail account and it asks me if I want to 'Save Changes' and I check Yes. owever, when I go back to that account in Mail/ Preferences, the box for Password is empty again. That doesn't help either.

I guess this is not unique to me, but I haven't found any solutions in other Forum questions.

Thanks for any suggestions.