Askbot for Laravel project

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In a way it's possible: deploy Askbot on your server perhaps "mounted" at some sub-url example.com/qa. If necessary it is possible to set up common authentication as authentication is shared via domain-specific cookies. Common authentication will require some custom coding - I'd start researching how a Django site can be integrated with Laravel - as there might be some code snippets/libraries that solve this, after that I'd look at the documentation/code of Django authentication.

Perhaps you could also read from Askbot database from the PHP after figuring out the database structure (although I'd keep Askbot and other components separate, if possible).

You'll end up with a system that runs a Django app and Laravel on the same server.

Another option - install Askbot on one server and give it a subdomain. This way you'll still be able to use common authentication cookies by setting the session cookie domain (in Django this variable is called SESSION_COOKIE_DOMAIN) to '.example.com'; the leading dot means that the cookie will be shared with all subdomains.