Burnse,
Yes, the LeMat carbine is real, original ones are extreamely rare, and very, very valuable, but it is a real gun, actually manufactured overseas for the Confederacy. Only a very few are known to exist, and of those, they all seem to be prototypes, with no two of them being quite alike. For example, I have seen photos of one rifle, nearly exactly like the one in the page you referanced, then another one, that was a pin fired model, and a third, that was a centerfire cartridge conversion model. It seems that the design hadn't been standardized before production was cancelled.
For a time, the F. LLI. Pietta company was developing a Lemat Reproduction Carbine to be carried in it's product line. For some reason they discontinued the project, either the carbine was proving to be just too costly to produce, or, as rumors have it, they backed off of developement because of legal concerns of chain fires severely injuring gun users.
The photograph in the web page you published looks to be one of the prototypes from F. LLI. Pietta, however, it is possible, some other manufacturer has taken up the project, and we may very well see a reproduction LeMat carbine made available to the general public.
As far as getting a "free" LeMat carbine is concerned, well good luck with that one, the originals are all in museums, and there has been no price established for them, however, the LeMat pistols typically sell at auction for upwards of $20,000.00. I would imagine that an original carbine would be upwards of $200,000.00.
Sincerely,
ElvinWarrior... aka... David, "EW"