rust under bathroom sink handle

it's only under the hot water handle and not the cold water handle. I removed the handle to have a look and here it is under the hot water:
http://www.flickr.com/photos/29161142@N02/7951373004/in/photostream
and here it is under the cold water looking how it's supposed to:
http://www.flickr.com/photos/29161142@N02/7951372760/in/photostream

Is there anything I can do to solve the problem?

Thanks, any reply greatly appreciated.

Yes buy a new faucet !

The odds of finding a replacement part there is slim.