I'd like to have Gecko, WebKit, or another webbrowser embedded in Java as a Swing/AWT control.
I'm looking for something different than JRex or JWebPane
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You could use JxBrowser. It features a Swing/JavaFX component that wraps the Chromium engine while providing a rich API and out-of-the-box hardware-acceleration through the GPU.
Unfortunately, they've dropped support for other engines (like Gecko and WebKit) since 4.0 version.
Note that it's not free, except for open-source projects.