… icon to start the project. … Any other ideas?
The best way to deploy a Java desktop application is using Java Web Start.
… JWS provides many appealing features including, but not limited to, splash screens, desktop integration, file associations, automatic update (including lazy downloads and programmatic control of updates), partitioning of natives & other resource downloads by platform, architecture or locale, configuration of run-time environment (minimum J2SE version, run-time options, RAM etc.), easy management of common resources using extensions..
Desktop integration is things like an icon on the desktop or menu items (in the start menu) to start the app. Each with customized icons.
At least on supported systems, AFAIU OS X will not allow any desktop icons to clutter the user interface and *nix machines typically do not support the menu items (or didn't last time I checked).