the backward payment without fees did not arrive even after a day. I searched the unconfirmed transactions on
http://bitcoincharts.com/bitcoin/but cannot find it. It also does not show up in my PC-client.
Is it possible that such small transactions without fee are dropped completely by the network?
When a transaction is made the bitcoin client announces the transaction to all the nodes it has peered with. Each node in turn announces it to its peers, and so on until eventually even the miner nodes know about it.
If for some reason the transaction didn't get announced properly, the client will re-send the transaction after some time has elapsed.
For best results, the sender should leave the Bitcoin client running until the transaction gets resent and makes it into a block. This could be many many hours if there are many "low priority" transactions queued, but at the current time there are few.