Hi Sebi,
I am wondering if in this new laptop you use a smartcards as the SSO is not compatible with smartcards as this is a passhare user only.
Also can you verify on both different machines with version 10.3.1 (working and not working ones) the following registry entry PGP_INSTALL_SSO.
( HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\PGP Corporation\PGP for x64 bits)
To be able to use SSO this value needs to be 1 there.
Look at the followig KB please:
Troubleshooting: Symantec Drive Encryption Single Sign-On
http://www.symantec.com/docs/TECH149470
From above KB please also verify solution 2,3,4 (although I agree that mostly this KB is for troubleshooting synchonization issue rather than just a SSO working or not issue)