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]
(..) - Javelin up! Javelin up! Countermeasures! No. I can see now that there are two men on top of the roof... ...and they're engaged in hand-to-hand combat. Shit! Yes! Skip Tyler is in! Walker, that last firewall just came down. (..)2013 White House Down