Finally, I've sent my Alix to an electronics repair shop. They just told me that one of the VT6105M NICs was dead. This fault caused my problem - after removing the NIC, my board boots again. However, I'm neither able to find this NIC in any shop in our country nor as a part of a used motherboard or a PCI card (people from the repair shop offered me a new NIC chip, however, it was very expensive).
I know that the NIC can be bought in China, but I still have a (possibly stupid) question - can be this VIA chip replaced by another "similar" one with same pinout from VIA Rhine series? Can it be replaced by a chip from another manufacturer?
Thanks for your responses. AWM