I have a Java application that places an icon in the unity system tray.

It shows up fine when I'm on a single monitor. When I have my laptop plugged into another monitor, there are two system trays: one for each screen.

The list of icons in the system tray are the same - except for my application's icon. It only shows up on my laptop's system tray, and not on my main monitor system tray.

In Java, I get a handle to the system tray via:
final SystemTray tray = SystemTray.getSystemTray();

I don't see an way to get the "other" system tray, if there really is one.

How would I go about getting my icon onto both, or alternatively, is there a way to only have the system tray on my main monitor?