I click on the icon SWRepublicCommando.exe, I tried the system compatibility mode (XP, Vista, 7), I marked the "run as administrator", but the game does not start. Have any of you had the exact same problem as me?
I have a Windows 8, Intel Core i5-3230 CPU 2,60 GHz, 4GB RAM, DirectX11, Intel HD Graphics 4000.
I ask for an explanation of my problem.

must be that intel Hd Graphics 4000 since its a integrated videocard.

This means that the game will not go on integrated graphics cards on a laptop?

I don't think it's the integrated graphics. I'm having the same issue using a GTX 680.

@OP: Is the game loading for you at all? Or is nothing happening at all when you try to start the game? If it's the latter, we might be in the same boat, and I have yet to find a fix for it, unfortunately.

I'm on a simple AspireOne 10.1" w/Windows 8.1 Home 64, the atom dual core processor w/integrated Intel graphics, and it runs great - it even has an option for my 1366 x 768 screen! :)
EDIT: Ooops! I spoke incorrectly here... While it does play fine on my 1366 x 768, I'm actually running it at 1280 x 960

Here's what I've done for getting all of my GOG games installing and working, when I previously could not:

It turned out to be an issue with my AV and/or my Firewall.

To simplify the entire process, I made a folder named "GOG" (it could be any name) where I move my GOG installer files to before installing. Then I just use the default location, which happens to be a single folder named GOG, or something like that, on my main hard drive.

Now I set both my AV and Firewall to let those folders just do their thing, however it may be worded in your system... it all works the same.

Before I did that, it would take about 90 million years for the installer to begin after double-clicking on it, and sometimes the games wouldn't want to play, even after setting up the permissions on the exe

The whole culprit was either anti-virus or my firewall, perhaps both. I no longer have any issues!

Good luck. It's not Win 8.1 with the latest updates. It works fine.