Thank you to all for the responses. I have looked at the info on the GOMOG site and several different (Poole and Rover) GEMS manuals and have not found much pertaining to this issue. I did find one reference to it that said to reset the GEMS using the Rovacom. It sounds like Colin did this and was unsuccessful over the long term.
So far this has not been a major problem, usually resolving itself by cycling ( off to start) the key 2 - 3 times. My main worry is it getting worse and being stranded somewhere along the road.
Just to clarify, I have not been setting the immobilizer when the car is parked. I only have one working PLIP and don't want to have the immobilizer set and that PLIP fail. The car battery was new when I purchased the car, with the problem occurring, and I keep the car on a battery maintainer. I will try the battery disconnect to see if that helps. I have had the battery disconnected in the past for short periods and have not had any additional problems. If it does disable the car, at least it will be at home and not on the road somewhere.
I will try the PLIP battery change procedure Andrew Green included and see if that brings the second PLIP back to life. The battery in that PLIP was probably original, but it still didn't work after a new battery.
A history on the car: It was delivered to the original owner in 2005 (after a 2 year wait). He drove the car for a short period (along with a Lotus) and then moved to Belgium in 2006. The car was stored, with the battery out, in San Diego, CA. The owner's daughter had the car gone through (at the San Diego Rover dealer) and put back in service in December of 2013, so the car sat for about 7 years with the battery removed. She could not find the removed battery, so she installed a new (same as original) one. The repair invoice indicates there was a code stored in the ECM for "battery voltage missing". They reported they found a bad connection and repaired it, and the code was cleared. This code has not reoccurred. I purchased the car in January of 2014, with 748 miles on it. Other than this problem, which comes and goes, and a failed rad fan relay, the car has been trouble free over about a thousand miles.
If anyone has any other ideas or can point me to someone familiar with this problem, please pass them on.
Many thanks,
Dave