Is it possible to search for existings auctions of a rare item?

This isn't possible and basically useless, because in case of magic items the item's names are generic (they always consist of the general item name and additional, randomized, terms that depend on the attributes, such as "Sword of Vampirism" or "Sword of the Leech" for a weapon with life-steal) and also, magic items with different values can have the same names.

In case of rare's it doesn't really make sense either, since there is a huge range of rare names for weapons with the same attributes and bonuses. If you look for a rare with a certain name, you might be missing out on a differently named item, that has better stats at the same cost.

Because of this, there is a specific option included in the Auction House search function to search for stats that you want to have on your item.

If you want an estimate off the amount of gold you can charge, you should better look for items with the same attributes as yours.

Rare item names are completely meaningless. They're completely random and don't have anything to do with the attributes available on the item (unlike magic items). But you can filter in the auction house by rarity, and then by attributes, in order to find an item that is similar to the one you are looking for (which will likely have a completely different name).

Patch 1.05 has made this much easier, by allowing you to just right-click an item in your inventory and choose to search for similar items. This will automatically fill in all the items attributes into the auction search interface. You can then remove or tweak values to find other items that are like yours.