Vodavi and LG are partners, actually, outside of North America I believe that all of their systems are marketed as the LG Telephone system - they even have a nifty hotel package off of the XTS.
I believe that the GDK 16 was their model of Mach 1. I have not heard of the LDK 24, but the LDK 100 is their version of the XTSc as far as I understand, and the LDK 300 is their XTS. Those all work with the "LDK" telephones....the same ones that work on our Mach 1, XTS, and XTSc...but theirs also come in a white color and a 2 flex button version.
Try calling these guys:
https://www.rdtcom.co.uk/Pages/contact.html they are in the UK, but they sell the whole range of LG telephones. They should be able to help you.
Disclaimer: I HAVE NEVER TALKED TO THEM, I only got their info from their web page, so I cannot vouch for their reputation.
Good luck!
Steve