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If I understand LionChief design, each controller will only run the locomotive it was designed for. Also if I am reading the literature properly, "... it works ALONG SIDE of Legacy/TMCC it does NOT work WITH Legacy/TMCC.", so that means you could get another controller for the 18079 and run it in conjunction with the LionChiefs. If that is what you mean by compatible then. "yes they are compatible".
I understand that the 18079 has a switch that the "Odyssey" part can be turned off.
LionChief runs at a constant track voltage (about 16 volts) as does Legacy/TMCC. The speed is controlled by the LionChief remote controlling the electronics in the LionChief locomotive. Or in the case of Legacy/TMCC the cab of the Legacy/TMCC system.
Legacy/TMCC will not (yet) control LionChief locomotives.
I had to correct my original post.
The locomotive you are looking at Lionel 18079 is a TMCC locomotive. You would need a Legacy system or a TMCC system to control it. As Iron Horsemen said they will work on the same layout because they both use a constant voltage, but you can't control LionChief with a Legacy/TMCC system. That is provide you are running AC voltage (LionChief can run on AC or DC)
If you are using AC You would need to control the LionChief locomotives with the LionChief remote and control the TMCC or Legacy locomotives with a Legacy/TMCC system.