Hosting and codeigniter

Hello, I have a hosting with a store in Codeigniter.

Separated to this now the client asked to add a system that I have developed but without the Framework involved. Can it be integrated in the same hosting?

Yes. You can have non-CI code alongside the CI framework, just need to make sure you're routing things correctly and you'll be fine (the htaccess should not route paths that resolve to files, while unknown route paths will resolve to CI - so if the file doesn't exist, CI takes over and routes to the controller if it exists).

If both have dynamic routing, this will require a complex htaccess file (where you manually setup the routing between the two systems so that /someCIcontroller/method routes to the codeigniter code and /yourOtherSystem/method routes to your custom code)