Find all integer solutions of the indefinite system of equations
$$\left\{\begin{array}{l}
x+y+z=3 \\
x^{3}+y^{3}+z^{3}=3
\end{array}\right.$$
3. Eliminate $z$ from the system of equations, to get
$$8-9 x-9 z+3 x^{2}+6 x y+3 y^{2}-x^{2} y-x y^{2}=0$$

Transform it into
$$8-3 x(3-x)-3 y(3-x)+x y(3-x)+y^{2}(3-x)=0,$$

which is $(3-x)\left(3 x+3 y-x y-y^{2}\right)=8$. Hence $(3-x) \mid 8$, so $3-x= \pm 1, \pm 2, \pm 4, \pm 8$, which means $x=-5,-1,1,2,4,5,7,11$. Substitute each value into the original system of equations to verify, and the complete set of integer solutions is $(x, y, z)=(1,1,1),(-5,4,4),(4,-5,4),(4,4,-5)$.