Winter sports vehicle rebuilt in each city (3,5)

I believe the answer is:

ice yacht

I'm a little stuck... Click here to teach me more about this clue!

'winter sports vehicle' is the definition.
The definition suggests a singular noun which matches the answer.

'rebuilt in each city' is the wordplay.
'each city' can be anagrammed to 'ICE YACHT'.
However, I am not sure how this is indicated.

Can you help me to learn more?

(Another definition for ice yacht that I've seen is " Winter vehicle".)

Want a hint initially instead of a full solution? Install my app