$$
\left\{\begin{array}{l}
x-y=29, \\
\sqrt[3]{x-1}-\sqrt[3]{y+2}=2 .
\end{array}\right.
$$

One, (20 points) Solve the system of equations
Let $\sqrt[3]{x-1}=a, \sqrt[3]{y+2}=b$, then the original system of equations becomes
$$
\left\{\begin{array}{l}
a^{3}-b^{3}=26, \\
a-b=2 .
\end{array}\right.
$$

From (1), we get
$$
a^{3}-b^{3}=(a-b)\left[(a-b)^{2}+3 a b\right]=26,
$$

which simplifies to $a b=3$.
Solving (2) and (3) together, we get $a=3, b=1$, or $a=-1$, $b=-3$. Substituting $\sqrt[3]{x-1}=a, \sqrt[3]{y+2}=b$, we get $x=28, y=1$ or $x=0, y=-29$.