try pLan instead of Hamachi.
in general, if the host has a public IP address, no programs are needed.
you can find pLan in my signature.

Zhenya, I understand that Hamachi and pLan have the same functions. Therefore, I don't want to bother with pLan yet, as I've already figured out Hamachi a bit in the last 2 days. I have Vista Ultimate at home, and my friend has XP 2 with Service Pack. I found this information on the internet:

"Go to the page where your network connections are displayed. In Vista, this is Network and Sharing Center - Manage network connections. In this window, find Advanced in the top menu (to make the top menu appear in Vista, press Alt in the network connections management window) and click on Advanced settings. Two windows will appear; in the top one, use the arrow to move Hamachi to the very top and click OK. Now, return to the Manage network connections page, right-click on the Hamachi connection, and go to properties. In the properties, go to TCP/IP settings. Enter the IP address assigned to you by the system; the subnet mask should be filled in automatically (255.0.0.0), as well as the gateway. If the gateway is not filled in, enter 5.0.0.1 there (if you have a different first digit, enter it). Now, at the very bottom of these properties, there is an Advanced button; click on it. Enter the gateway there and set the metric to automatically, and at the bottom, check the box to set it automatically. That's it. If you use Hamachi for games, remember that many assume you are on the same network as the server. To bypass this problem, in Hamachi, go to the properties, on the Window tab, and check the box to show additional menu items. Then, right-click on the server, go to the Advanced menu, and assign it an alias in the peer VPN alias field. Do not change the address, but assign an alias."

I did as it was written there. But I'm not sure if this will help; I need to try...
P.S. Zhenya. My friend and I have internet connected at home via twisted pair without any modems or routers. In our "village," this is called internet through a network, but our networks are different. Does this mean that we have a static public IP address and can play without any programs? Are there any other ways to fix this problem? :(
P.P.S. Yesterday, Heroes identified my IP as 80.245.122.78, and today as 80.245.122.65. And also, in the Hamachi instructions, it says that when Heroes starts this program and creates a network, it should identify the IP assigned by Hamachi (mine is 5.2.229.229), but it identifies the previous one...

Added after 7 hours 54 minutes
The situation has changed slightly. I configured Hamachi so that Heroes identifies the IP of this program. It shows when my friend creates the game, but I can't connect to him; it says: "Sorry, an error occurred while connecting to this session. Please try again." I tried to connect with pLan, but we can't connect either via TCP/IP or direct connection, although when we go into the game via LAN, we can see the host and play (although there are lags). I would like to ask if any kind-hearted people could give me their ICQ number. I would like to discuss this in real-time.