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]
(..) Have you stopped stealing bread rolls? Ballet girl, Oriental type. How can I help it? How is... How are the Shcherbatskys? You mean Kitty. -Is she engaged now? -No. Vronsky went back to St. Petersburg. Huh. Kitty will be seeing Dolly at Ergoshovo in the summer. (..)2012 Anna Karenina