If you muse on something, you think about it, usually saying or writing what you are thinking at the same time.
e.g. Many of the papers muse on the fate of the President...
e.g. 'As a whole,' she muses, 'the 'organized church' turns me off'...
His musings were interrupted by Montagu who came and sat down next to him.
A muse is a person, usually a woman, who gives someone, usually a man, a desire to create art, poetry, or music, and gives them ideas for it.
e.g. Once she was a nude model and muse to French artist Henri Matisse.
e.g. ...the muses who fuel his inspiration.