hello, for the variable $a$ i got the following equation $(112a^6-80a^3+49)(448a^6-536a^3+169)(a-1)(a^2+a+1)(2a+1)(1-2a+4a^2)=0$ Sonnhard.

we can write that $(a-b)((a-b)^{2}+3c^{2})=0$ (1)-(2) similar have 2 equation. hence a=b=c then find a,b,c don't difficult.

The solution is $(a,b,c) = (1,1,1)$