He won't be able to attend because he's 'otherwise engaged'. [=he is busy doing something else]
('US') He is 'engaged' in research. = ('Brit') He is 'engaged' on research.
She 'engaged' him 'in' conversation. [=she started a conversation with him]
(..) This is a quality expected in every Starfleet captain. Excuse me, sir. We've received a distress call from Vulcan. With our primary fleet engaged in the Laurentian system, I hereby order all cadets to report to hangar immediately. Dismissed. - Who was that pointy-eared bastard? (..)2009 Star Trek
(..) fight them off. George, I can't do this without you. OK, I need you to push now! SYSTEMS FAILING COLLISION COURSE ENGAGED - What is it? - It's a boy. A boy? - Tell me about him. - He's beautiful. (..)2009 Star Trek