$\displaystyle \left\{ \begin{array}{l}a^{5} = 5b^{3} - 4c \\
b^{5} = 5c^{3} - 4a \\
c^{5} = 5a^{3} - 4b\end{array} \right\}$
If you add these equations you get:
(a-2)(a-1)a(a+1)(a+2) + (b-2)(b-1)b(b+1)(b-2) + (c-2)(c-1)c(c+1)(d+1) = 0
One solution is: a=b=c=2 or 1 or 0 or -1 or -2
I don't know if there are other solutions, maybe someone can proof that a,b,c must be equal?