Are you sure that the mystery application is not simply using the hostname gateway.2wire.net instead of a hard coded IP address? If you are using the 2wire router for DNS resolution, that hostname will always resolve to the current IP address for the router.
Of course since you refuse to provide any details, any answer will simply be a wild guess.--
We can never have enough of nature.
We need to witness our own limits transgressed, and some life pasturing freely where we never wander.
Test your firewall.