1. Subtract the second equation from the first:

$x^{2}-2 x+y^{4}-8 y^{3}+24 y^{2}-32 y=-17 \rightarrow(x-1)^{2}+(y-2)^{4}=0 \rightarrow\left\{\begin{array}{l}x=1 \\ y=2\end{array}\right.$

Then $z=x^{2}+y^{4}-8 y^{3}=1+16-64=-47$
Answer: the only solution is $x=1, y=2, z=-47$.