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If we define invariant momentum like invariant mass at v=0 then we get that invariant momentum - the momentum at zero speed - is 0. I suppose that that is why there is no invariant momentum...True, I'm not asking about the frame of a certain particle properties.

It is hard to explain, I'm referring to the concept that the quantity is invariant regardless the state of motion.

Take momentum instead of energy, there is no invariant momentum. So it might sound somewhat confusing that there is an energy invariant when in SR what is conserved is the momentum-energy.