I want ot get internet wireless but the router I have it's not on the list nintendo has.
The list of compatible routers only lists routers that Nintendo has tested and can confirm work. However, Nintendo cannot test all the routers out there.
Almost any router will work. The best way to find out is to try it!
Just be sure that you do not have WPA security enabled.
You can use WPA security if you only need to connect a Wii.
If you have a DS/DSi/DSiXL/3DS and you want to be able to play ANY DS game over the internet, then you are correct; WPA cannot be used.
Nintendo has some networks, but they're private whatsoever. You can connnect the connection to the networks that your computer has with just a password to the network and the router.