For log-in integration you will need to set up a sso-handler or login handler (ie. write some php code in an extension). You will find some existing extensions that you can use as an example.
For integrating external applications, you will need to create a new module and view inside an extension. You will most likely need to write some code that hacks the received urls in ezp form to translate them into the form your app expects. The view will then include the php code from your application using ob_start and ob_end and set it into the $Result['content'] variable. You can decide whether you want to have the default eZ Publish pagelayout template around your app or not. In any case, the ezpublish policies will apply - to allow access to your smarty outputes the user will need to have a proper eZ permission granted.
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