Case demanding investigation at first during dance (6)
I believe the answer is:
valise
'case' is the definition.
(valise is a kind of case)
'investigation at first during dance' is the wordplay.
I cannot really see how this works, but
'first' could be 'i' (I in Roman numerals eg Elizabeth I) and 'i' is found within the answer.
'dance' could be 'valse' (valse is a kind of dance) and 'valse' is found in the remaining letters.
This accounts for all the letters.
This explanation may well be incorrect...
'demanding' is the link.
Can you help me to learn more?
(Other definitions for valise that I've seen before include "Elvis A? (anag)" , "Small overnight bag" , "Overnight case" , "A small suitcase for Leavis" , "Small travelling bag or suitcase" .)