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I used LimeWire version 4.18.8 fine with Leopard on iMac. I got new iMac with Snow Leopard and when I search there is not a single result comes out (I test on such popular name as B**tles). I don't want to update to a new version of LimeWire because of some search limitations there. Can 4.18.8 work with Snow Leopard? or what should I do to make it work?
Please help.

Yes LW 4 will work with Snow Leopard. However, this is an issue for LW 4. When this happens in future, quit LW, wait a few mins & re-open. My guess is your ultrapeers are not responding properly to you.
The alternative is to go to Connections window, remove the longest connected Ultrapeer, wait until LW finds a replacement UP & the connection settles. Then try searches again. Repeat this process. ;)

Also, try searching something else. Sometimes searching same term too many times can result in ultrapeers starting to ignore the search query. So change your search term or technique. ;)

Generally when faced with your scenario, this usually only lasts for 1-5 days. I've experienced this issue myself whilst using LW 4.

Sorry for the very late reply btw, not sure how your question escaped us. :o