Where can I find examples of cross-site authorization?

Question

PHP

Good day.
There are 4 sites available, one of them is the main one.
2 of them are affiliated, so I can't give direct access to my database.
It is necessary to implement authorization on sites through one account on the main site.
As in VK, you have a profile, you can register through it on the sites that made this possible.
I read about the OpenID and oAuth2 system, but I couldn’t find anywhere how to deploy it on ready-made sites (not frameworks, not CMS). Everywhere I find only the client setting, but I can not find the server itself. I've never worked with APIs and stuff before, but you always have to start somewhere.
Help me find exactly not a theory, but where in practice the deployment of the server and client is shown, without using any frameworks.
The task is one, to authorize partner sites through the main site, so access levels and other things do not play a role.