Moved the thread.
We will break it down then
Panasonic. Company name but does give sense to the letters to follow
VL- Panasonic's code for Lithium rechargeable batteries
2020 gets broken down twice more and is reasonably standard across most coin/button cell manufacturers/vendors.
20 is a rounded down diameter in mm (20 in this case but 24 often means 24.5)
the final 20 is the thickness in mm if you put a decimal point between the two numbers (2.0mm in this case)
The bigger trick is finding one with tags welded on in the proper arrangement. You could get fancy with the hot glue and wires I guess but tags are preferable.
Being size values if you can find something of the same voltage that fits in there and has a reasonable charge capacity (or your are willing to stick it in the linker to charge more often) then by all means solder it in.
I did see some of these batteries, even with the tags in a somewhat suitable arrangement, got used in a car remote (some BMW if memory serves) which pumped the price up on the likes of ebay and amazon (people will pay something silly like $15 to get their key fob, or even keys in general, working again if the alternative is $70 from a main dealer) but does mean you can get them from non electronics suppliers these days.
Also you do care about recharge for this -- when a flash cart is just holding SRAM saves it can last for ages, when it is also doing real time clock it is more troubling. With that said it should be noted that the EZ2 has no specialist charge management* on it beyond a diode so if your battery can handle a constant input of the voltage in question (entirely possible) and charge/not die/not blow up in your face then go for it.
*battery chemistry and charge management is a complex and boring subject, though
eevblog 176 is a nice intro to an aspect of it all if you are so inclined.
http://industrial.panasonic.com/www-data/pdf/AAA4000/AAA4000PE16.pdf being the one for the battery in question.
"VL2020-1HFE" and "VL2020-1VCE" just for the sake of completeness refer to the types of welded tags on them, neither would seem to be the type of tags used on the EZ4, the 1VCE being the closest (for forum searchers it is also called the VL-2020/VCN in some places). but the tags will be spread apart compared to the normal EZ2 so you will have to solder a jumper wire. I can not seem to the find the relevant one on Panasonic's website right now, however I am not looking at any retired models. Annoyingly for you (and likely me if I bother to replace mine any time in the future) is the smaller sizes and bigger sizes come with the sort of tags you want for this (
http://www.digikey.com/product-sear...ee=0&rohs=0&quantity=&ptm=0&fid=0&pageSize=50 ). Capacity wise it is about half but that should still be reasonable.