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In short, after three sleepless nights, I finally figured out how to fix this bug when playing under Windows 7. The key to this is the input_a2.cfg configuration file, which is located in the user folder, specifically C:\Users\your_username\Documents\My Games\Heroes of Might and Magic V — Tribes of the East\Profiles\Your_in_game_username. The problem is that when launching the game in Windows 7, you cannot change the graphics settings in the game itself; you can only do it through the corresponding configuration file using Notepad, the path to which I indicated above. You need to open this file with Notepad. Attention! Before editing this file, you need to set the graphics settings in the game by clicking the "Default" button in the corresponding section of the game, and only then edit this file.
The graphics settings parameters in the file have names like:
setvar gfx_16bit_mode = 0
setvar gfx_16bit_textures = 0
setvar gfx_anisotropic_filter = 8
and so on. Attention! The easiest way to set the desired graphics level (the game defaults to low settings) is to make a copy of the configuration file, then go into the game and set the desired settings there. Then exit the game, open both files - the one you copied and the one that was just edited by the game - and compare the changed parameter values. Next, we simply set the desired parameter values in the file we copied and save it to replace the file that was edited by the game, after closing the latter.

After all these manipulations, the textures in the main menu remained blurry for me, BUT all other textures started to display correctly.
The graphics settings parameters in the file have names like:
setvar gfx_16bit_mode = 0
setvar gfx_16bit_textures = 0
setvar gfx_anisotropic_filter = 8
and so on. Attention! The easiest way to set the desired graphics level (the game defaults to low settings) is to make a copy of the configuration file, then go into the game and set the desired settings there. Then exit the game, open both files - the one you copied and the one that was just edited by the game - and compare the changed parameter values. Next, we simply set the desired parameter values in the file we copied and save it to replace the file that was edited by the game, after closing the latter.
After all these manipulations, the textures in the main menu remained blurry for me, BUT all other textures started to display correctly.