Firefox can't find JRE when installed to non-default location

FULL PRODUCT VERSION :
java version "1.6.0_31"
Java(TM) SE Runtime Environment (build 1.6.0_31-b05)
Java HotSpot(TM) Client VM (build 20.6-b01, mixed mode, sharing)

ADDITIONAL OS VERSION INFORMATION :
Microsoft Windows XP [Version 5.1.2600]

EXTRA RELEVANT SYSTEM CONFIGURATION :
Firefox 10.0.1

A DESCRIPTION OF THE PROBLEM :
I have installed different JRE versions on my sytsem (JRE 1.6.0_27, JRE 1.6.0_30).
I installed the new JRE version in a non-default directory. After successful instaalation I could see parallel installation of JRE 1.6.0_27 and 1.6.0_31.

After vstarting firefox i was asked to install a new java plugin. I agreed and could work with this new java version.
After installation of a new JDK 1.6.0_31 and rebooting my system, Firefox was'nt able to find a java plugin.
The java test page http://www.java.com/de/download/testjava.jsp told me that there was no java plugin available, the add-on/plugin list really shows no java any more.
I had to deinstall and install my JRE 1.6.0_31 again (default installation)

Maybe there is a correlation with your firefox bug fix in JRE 1.6.0_31.

REGRESSION. Last worked in version 6u29

STEPS TO FOLLOW TO REPRODUCE THE PROBLEM :
1) multiple installation of JRE 1.6.0_xx
2) install JRE 1.6.0_31 in a non-default directory
3) start firefox -> install plugin
4) install JDK 1.6.0_31 (maybe this step is not necessary)
5) reboot your system (maxbe this step is not necessary)
6) start firefox -> no java plugin is found any more

EXPECTED VERSUS ACTUAL BEHAVIOR :
EXPECTED -
newest java plugin should be found
ACTUAL -
no java plugin is installed

REPRODUCIBILITY :
This bug can be reproduced always.

CUSTOMER SUBMITTED WORKAROUND :
default installation of JRE (no parallel JREs)